# RD Sharma Solutions for Class 7 Maths Chapter 15 Properties of Triangles

The PDF of RD Sharma Solutions for Class 7 Maths Chapter 15 Properties of Triangles are downloaded from the given links. The solutions are prepared by BYJUâ€™S experts in Maths. The RD Sharma Solutions for exercise wise problems are prepared after vast research is conducted on each topic. The step-wise solutions prepared by subject experts help students to solve problems with competence.

Chapter 15, Properties of Triangles includes five exercises. RD Sharma Solutions for Class 7 includes answers to all questions present in these exercises. Let us have a look at some of the concepts that are being discussed in this Chapter.

• Definition and meaning of triangles
• Sides of the triangle
• Angles of the triangle
• Elements or parts of the triangle
• Interior and exterior angle of the triangle
• Triangular region
• Types of triangle by considering the lengths of their sides
• Scalene triangle
• Isosceles triangle
• Equilateral triangle
• Types of triangle by considering the measures of their angles
• Acute triangle
• Right triangle
• Obtuse triangle
• Perimeter of triangle
• Angle sum property of triangle
• Exterior and interior opposite angles of a triangle
• Triangle inequality property
• Pythagoras theorem
• Converse of Pythagoras theorem
• Pythagorean triplet

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Exercise 15.1 Page No: 15.4

1. Take three non-collinear points A. B and C on a page of your notebook. Join AB, BC and CA. What figure do you get? Name the triangle. Also, name

(i) The side opposite to âˆ B

(ii) The angle opposite to side AB

(iii) The vertex opposite to side BC

(iv) The side opposite to vertex B.

Solution:

(i) The side opposite to âˆ B is AC

(ii) The angle opposite to side AB is âˆ C

(iii) The vertex opposite to side BC is A

(iv) The side opposite to vertex B is AC

2. Take three collinear points A, B and C on a page of your note book. Join AB. BC and CA. Is the figure a triangle? If not, why?

Solution:

No, the figure is not a triangle. By definition a triangle is a plane figure formed by three non-parallel line segments

3. Distinguish between a triangle and its triangular region.

Solution:

Triangle:

A triangle is a plane figure formed by three non-parallel line segments.

Triangular region:

Whereas, itâ€™s triangular region includes the interior of the triangle along with the triangle itself.

4. D is a point on side BC of aÂ â–³CAD is joined. Name all the triangles that you can observe in the figure. How many are they?

Solution:

We can observe the following three triangles in the given figure

â–³ ABC

â–³ ACD

5. A, B, C and D are four points, and no three points are collinear. AC and BD intersect at O. There are eight triangles that you can observe. Name all the triangles

Solution:

Given A, B, C and D are four points, and no three points are collinear

â–³ ABC

â–³ ACD

â–³ DBC

â–³ ABD

â–³ AOB

â–³ BOC

â–³ COD

â–³ AOD

6. What is the difference between a triangle and triangular region?

Solution:

Triangle:

A triangle is a plane figure formed by three non-parallel line segments.

Triangular region:

Whereas, itâ€™s triangular region includes the interior of the triangle along with the triangle itself.

7. Explain the following terms:

(i) Triangle

(a) Parts or elements of a triangle

(iii) Scalene triangle

(iv) Isosceles triangle

(v) Equilateral triangle

(vi) Acute triangle

(vii) Right triangle

(viii) Obtuse triangle

(ix) Interior of a triangle

(x) Exterior of a triangle

Solution:

(i) A triangle is a plane figure formed by three non-parallel line segments.

(ii) The three sides and the three angles of a triangle are together known as the parts or elements of that triangle.

(iii) A scalene triangle is a triangle in which no two sides are equal.

(iv) An isosceles triangle is a triangle in which two sides are equal.

(v) An equilateral triangle is a triangle in which all three sides are equal.

(vi) An acute triangle is a triangle in which all the angles are less than 90o.

(vii) A right angled triangle is a triangle in which one angle should be equal to 90o.

(viii) An obtuse triangle is a triangle in which one angle is more than 90o.

(ix) Â The interior of a triangle is made up of all such points that are enclosed within the triangle.

(x) The exterior of a triangle is made up of all such points that are not enclosed within the triangle.

8. In Fig. 11, the length (in cm) of each side has been indicated along the side. State for each triangle angle whether it is scalene, isosceles or equilateral:

Solution:

(i) The given triangle is a scalene triangle because no two sides are equal.

(ii) The given triangle is an isosceles triangle because two of its sides, viz. PQ and PR, are equal.

(iii) The given triangle is an equilateral triangle because all its three sides are equal.

(iv) The given triangle is a scalene triangle because no two sides are equal.

(v) The given triangle is an isosceles triangle because two of its sides are equal.

9. In Fig. 12, there are five triangles. The measures of some of their angles have been indicated. State for each triangle whether it is acute, right or obtuse.

Solution:

(i) The given triangle is a right triangle because one of its angles is 90o.

(ii) The given triangle is an obtuse triangle because one of its angles is 120o, which is greater than 90o

(iii) The given triangle is an acute triangle because all its angles are less than 90o

(iv) The given triangle is a right triangle because one of its angle is 90o.

(v) The given triangle is an obtuse triangle because one of its angle is 120o, which is greater than 90o.

10. Fill in the blanks with the correct word/symbol to make it a true statement:

(i) A triangle has â€¦â€¦. sides.

(ii) A triangle has â€¦â€¦..vertices.

(iii) A triangle has â€¦â€¦..angles.

(iv) A triangle has â€¦â€¦â€¦parts.

(v) A triangle whose no two sides are equal is known as â€¦â€¦â€¦

(vi) A triangle whose two sides are equal is known as â€¦â€¦â€¦.Â

(vii) A triangle whose all the sides are equal is known as â€¦â€¦â€¦

(viii) A triangle whose one angle is a right angle is known as â€¦â€¦..

(ix) A triangle whose all the angles are of measure less than 90′ is known as â€¦â€¦â€¦

(x) A triangle whose one angle is more than 90′ is known as â€¦â€¦â€¦.

Solution:

(i) Three

(ii) Three

(iii) Three

(iv) Six

(v) A scalene triangle

(vi) An isosceles triangle

(vii) An equilateral triangle

(viii) A right triangle

(ix) An acute triangle

(x) An obtuse triangle

11. In each of the following, state if the statement is true (T) or false (F):

(i) A triangle has three sides.

(ii) A triangle may have four vertices.

(iii) Any three line-segments make up a triangle.

(iv) The interior of a triangle includes its vertices.

(v) The triangular region includes the vertices of the corresponding triangle.

(vi) The vertices of a triangle are three collinear points.

(vii) An equilateral triangle is isosceles also.

(viii) Every right triangle is scalene.

(ix) Each acute triangle is equilateral.

(x) No isosceles triangle is obtuse.

Solution:

(i) True

(ii) False

Explanation:

Any three non-parallel line segments can make up a triangle.

(iii) False.

Explanation:

Any three non-parallel line segments can make up a triangle.

(iv) False.

Explanation:

The interior of a triangle is the region enclosed by the triangle and the vertices are not enclosed by the triangle.

(v) True.

Explanation:

The triangular region includes the interior region and the triangle itself.

(vi) False.

Explanation:

The vertices of a triangle are three non-collinear points.

(vii) True.

Explanation:

In an equilateral triangle, any two sides are equal.

(viii) False.

Explanation:

A right triangle can also be an isosceles triangle.

(ix) False.

Explanation:

Each acute triangle is not an equilateral triangle, but each equilateral triangle is an acute triangle.

(x) False.

Explanation:

An isosceles triangle can be an obtuse triangle, a right triangle or an acute triangle

Exercise 15.2 Page No: 15.12

1. Two angles of a triangle are of measures 150o and 30o. Find the measure of the third angle.

Solution:

Given two angles of a triangle are of measures 150o and 30o

Let the required third angle be x

We know that sum of all the angles of a triangle = 180o

150o + 30o + x = 180o

135o + x = 180o

x = 180o â€“ 135o

x = 45o

Therefore the third angle is 45o

2. One of the angles of a triangle is 130o, and the other two angles are equal. What is the measure of each of these equal angles?

Solution:

Given one of the angles of a triangle is 130o

Also given that remaining two angles are equal

So let the second and third angle be x

We know that sum of all the angles of a triangle = 180o

130o + x + x = 180o

130o + 2x = 180o

2x = 180o â€“ 130o

2x = 50o

x = 50/2

x = 25o

Therefore the two other angles are 25o each

3. The three angles of a triangle are equal to one another. What is the measure of each of the angles?

Solution:

Given that three angles of a triangle are equal to one another

So let the each angle be x

We know that sum of all the angles of a triangle = 180o

x + x + x = 180o

3x = 180o

x = 180/3

x = 60o

Therefore angle is 60o each

4. If the angles of a triangle are in the ratio 1: 2: 3, determine three angles.

Solution:

Given angles of the triangle are in the ratio 1: 2: 3

So take first angle as x, second angle as 2x and third angle as 3x

We know that sum of all the angles of a triangle = 180o

x + 2x + 3x = 180o

6x = 180o

x = 180/6

x = 30o

2x = 30o Ã— 2 = 60o

3x = 30o Ã— 3 = 90o

Therefore the first angle is 30o, second angle is 60o and third angle is 90o.

5. The angles of a triangle are (x âˆ’ 40)o, (x âˆ’ 20)o and (1/2 âˆ’ 10)o. Find the value of x.

Solution:

Given the angles of a triangle are (x âˆ’ 40)o, (x âˆ’ 20)o and (1/2 âˆ’ 10)o.

We know that sum of all the angles of a triangle = 180o

(x âˆ’ 40)o + (x âˆ’ 20)o + (1/2 âˆ’ 10)o = 180o

x + x + (1/2) â€“ 40o â€“ 20o â€“ 10o = 180o

x + x + (1/2) â€“ 70o = 180o

(5x/2) = 180o + 70o

(5x/2) = 250o

x = (2/5) Ã— 250o

x = 100o

Hence the value of x is 100o

6. The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10o. Find the three angles.

Solution:

Given that angles of a triangle are arranged in ascending order of magnitude

Also given that difference between two consecutive angles is 10o

Let the first angle be x

Second angle be x + 10o

Third angle be x + 10o + 10o

We know that sum of all the angles of a triangle = 180o

x + x + 10o + x + 10o +10o = 180o

3x + 30 = 180

3x = 180 – 30

3x = 150

x = 150/3

x = 50o

First angle is 50o

Second angle x + 10o = 50 + 10 = 60o

Third angle x + 10o +10o = 50 + 10 + 10 = 70o

7. Two angles of a triangle are equal and the third angle is greater than each of those angles by 30o. Determine all the angles of the triangle

Solution:

Given that two angles of a triangle are equal

Let the first and second angle be x

Also given that third angle is greater than each of those angles by 30o

Therefore the third angle is greater than the first and second by 30o = x + 30o

The first and the second angles are equal

We know that sum of all the angles of a triangle = 180o

x + x + x + 30o = 180o

3x + 30 = 180

3x = 180 – 30

3x = 150

x = 150/3

x = 50o

Third angle = x + 30o = 50o + 30o = 80o

The first and the second angle is 50o and the third angle is 80o.

8. If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right triangle.

Solution:

Given that one angle of a triangle is equal to the sum of the other two

Let the measure of angles be x, y, z

Therefore we can write above statement as x = y + z

x + y + z = 180o

Substituting the above value we get

x + x = 180o

2x = 180o

x = 180/2

x = 90o

If one angle is 90o then the given triangle is a right angled triangle

9. If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled.

Solution:

Given that each angle of a triangle is less than the sum of the other two

Let the measure of angles be x, y and z

From the above statement we can write as

x > y + z

y < x + z

z < x + y

Therefore triangle is an acute triangle

10. In each of the following, the measures of three angles are given. State in which cases the angles can possibly be those of a triangle:

(i) 63o, 37o, 80o

(ii) 45o, 61o, 73o

(iii) 59o, 72o, 61o

(iv) 45o, 45o, 90o

(v) 30o, 20o, 125o

Solution:

(i) 63o + 37o + 80o = 180o

Angles form a triangle

(ii) 45o, 61o, 73o is not equal to 180o

Therefore not a triangle

(iii) 59o, 72o, 61o is not equal to 1800

Therefore not a triangle

(iv) 45o+ 45o+ 90o = 180o

Angles form a triangle

(v) 30o, 20o, 125o is not equal to 180o

Therefore not a triangle

11. The angles of a triangle are in the ratio 3: 4: 5. Find the smallest angle

Solution:

GivenÂ that angles of a triangle are in the ratio: 3: 4: 5

Therefore let the measure of the angles be 3x, 4x, 5x

We know that sum ofÂ the angles of a triangle =180o

3x + 4x + 5x = 180o

12x = 180o

x = 180/12

x = 15o

Smallest angle = 3x

= 3 Ã— 15o

= 45o

Therefore smallest angle = 45o

12. Two acute angles of a right triangle are equal. Find the two angles.

Solution:

Given that acute angles of a right angled triangle are equal

We know that Right triangle: whose one of the angle is a right angle

Let the measure of angle be x, x, 90o

x + x + 90o= 180o

2x = 90o

x = 90/2

x = 45o

The two angles are 45o and 45o

13. One angle of a triangle is greater than the sum of the other two. What can you say about the measure of this angle? What type of a triangle is this?

Solution:

Given one angle of a triangle is greater than the sum of the other two

Let the measure of the angles be x, y, z

From the question we can write as

x > y + zÂ or

y > x + zÂ Â  or

z > x + y

x or y or z > 90o which is obtuse

Therefore triangle is an obtuse angle

14. In the six cornered figure, (fig. 20), AC, AD and AE are joined. Find âˆ FAB + âˆ ABC + âˆ BCD + âˆ CDE + âˆ DEF + âˆ EFA.

Solution:

We know that sum of the angles of a triangle is 180o

Therefore in âˆ†ABC, we have

âˆ CAB + âˆ ABC + âˆ BCA = 180o â€¦â€¦.. (i)

Inâ–³ ACD, we have

âˆ DAC + âˆ ACD + âˆ CDA = 180o â€¦â€¦.. (ii)

In â–³AEF, we have

âˆ FAE + âˆ AEF + âˆ EFA = 180o â€¦â€¦â€¦. (iv)

Adding (i), (ii), (iii), (iv) we get

âˆ CAB + âˆ ABC + âˆ BCA + âˆ DAC + âˆ ACD + âˆ CDA + âˆ EAD + âˆ ADE + âˆ DEA + âˆ FAE + âˆ AEF +âˆ EFA = 720o

Therefore âˆ FAB + âˆ ABC + âˆ BCD + âˆ CDE + âˆ DEF + âˆ EFA = 720o

15. Find x, y, z (whichever is required) from the figures (Fig. 21) given below:

Solution:

(i) In â–³ABC and â–³ADE we have,

âˆ  ADE = âˆ  ABC [corresponding angles]

x = 40o

âˆ AED = âˆ ACB (corresponding angles)

y = 30o

We know that the sum of all the three angles of a triangle is equal to 180o

x + y + z = 180o (Angles of â–³ADE)

Which means: 40o + 30o + z = 180o

z = 180o – 70o

z = 110o

Therefore, we can conclude that the three angles of the given triangle are 40o, 30oÂ and 110o

(ii) Â We can see that in â–³ADC, âˆ ADC is equal to 90o.

We also know that the sum of all the angles of a triangle is equal to 180o.

Which means: 45o + 90o + y = 180o (Sum of the angles of â–³ADC)

135o + y = 180o

y = 180o â€“ 135o.

y = 45o.

We can also say that in â–³ ABC, âˆ ABC + âˆ ACB + âˆ BACÂ is equal to 180o.

(Sum of the angles of â–³ABC)

40o + y + (x + 45o) = 180o

40o + 45o + x + 45o = 180oÂ (y = 45o)

x = 180o â€“130o

x = 50o

Therefore, we can say that the required angles are 45o and 50o.

(iii) We know that the sum of all the angles of a triangle is equal to 180o.

Therefore, for â–³ABD:

âˆ ABD +âˆ ADB + âˆ BAD = 180Â° (Sum of the angles of â–³ABD)

50o + x + 50o = 180o

100o + x = 180o

x = 180o â€“ 100o

x = 80o

For â–³ ABC:

âˆ ABC + âˆ ACB + âˆ BAC = 180o (Sum of the angles of â–³ABC)

50o + z + (50o + 30o) = 180o

50o + z + 50o + 30o = 180o

z = 180o â€“ 130o

z = 50o

Using the same argument for â–³ADC:

âˆ ADC + âˆ ACD + âˆ DAC = 180o (Sum of the angles of â–³ADC)

y + z + 30o = 180o

y + 50o + 30o = 180o (z = 50o)

y = 180o â€“ 80o

y = 1000

Therefore, we can conclude that the required angles are 80o, 50o and 100o.

(iv) In â–³ABC and â–³ADE we have:

âˆ ADE = âˆ ABC (Corresponding angles)

y = 50o

Also, âˆ AED = âˆ ACBÂ (Corresponding angles)

z = 40o

We know that the sum of all the three angles of a triangle is equal to 180o.

We can write as x + 50o + 40o = 180o (Angles of â–³ADE)

x = 180o â€“ 90o

x = 90o

Therefore, we can conclude that the required angles are 50o, 40o and 90o.

16. If one angle of a triangle is 60o and the other two angles are in the ratio 1: 2, find the angles.

Solution:

Given that one of the angles of the given triangle is 60o.

Also given that the other two angles of the triangle are in the ratio 1: 2.

Let one of the other two angles be x.

Therefore, the second one will be 2x.

We know that the sum of all the three angles of a triangle is equal to 180o.

60o + x + 2x = 180o

3x = 180o â€“ 60o

3x = 120o

x = 120o/3

x = 40o

2x = 2 Ã— 40o

2x = 80o

Hence, we can conclude that the required angles are 40o and 80o.

17. If one angle of a triangle is 100o and the other two angles are in the ratio 2: 3. Find the angles.

Solution:

Given that one of the angles of the given triangle is 100o.

Also given that the other two angles are in the ratio 2: 3.

Let one of the other two angle be 2x.

Therefore, the second angle will be 3x.

We know that the sum of all three angles of a triangle is 180o.

100o + 2x + 3x = 180o

5x = 180o â€“ 100o

5x = 80o

x = 80/5

x = 16

2x = 2 Ã—16

2x = 32o

3x = 3Ã—16

3x = 48o

Thus, the required angles are 32o and 48o.

18. In â–³ABC, if 3âˆ A = 4âˆ B = 6âˆ C, calculate the angles.

Solution:

We know that for the given triangle, 3âˆ A = 6âˆ C

âˆ A = 2âˆ C â€¦â€¦. (i)

We also know that for the same triangle, 4âˆ B = 6âˆ C

âˆ B = (6/4) âˆ C â€¦â€¦ (ii)

We know that the sum of all three angles of a triangle is 180o.

Therefore, we can say that:

âˆ A + âˆ B + âˆ C = 180o (Angles of â–³ABC)â€¦â€¦ (iii)

On putting the values of âˆ A and âˆ B in equation (iii), we get:

2âˆ C + (6/4) âˆ C + âˆ C = 180o

(18/4)Â âˆ C = 180o

âˆ C = 40o

From equation (i), we have:

âˆ A = 2âˆ C = 2 Ã— 40

âˆ A = 80o

From equation (ii), we have:

âˆ B = (6/4) âˆ C = (6/4) Ã— 40o

âˆ B = 60o

âˆ A = 80o, âˆ B = 60o, âˆ C = 40o

Therefore, the three angles of the given triangle are 80o, 60o, and 40o.

19. Is it possible to have a triangle, in which

(i) Two of the angles are right?

(ii) Two of the angles are obtuse?

(iii) Two of the angles are acute?

(iv) Each angle is less than 60o?

(v) Each angle is greater than 60o?

(vi) Each angle is equal to 60o?

Solution:

(i) No, because if there are two right angles in a triangle, then the third angle of the triangle must be zero, which is not possible.

(ii) No, because as we know that the sum of all three angles of a triangle is always 180o. If there are two obtuse angles, then their sum will be more than 180o, which is not possible in case of a triangle.

(iii) Yes, in right triangles and acute triangles, it is possible to have two acute angles.

(iv) No, because if each angle is less than 60o, then the sum of all three angles will be less than 180o, which is not possible in case of a triangle.

(v) No, because if each angle is greater than 600, then the sum of all three angles will be greater than 1800, which is not possible.

(vi) Yes, if each angle of the triangle is equal to 600, then the sum of all three angles will be 1800, which is possible in case of a triangle.

20. Inâ–³ ABC, âˆ A = 100o, AD bisects âˆ A and AD âŠ¥ BC. Find âˆ B

Solution:

Given that in â–³ABC, âˆ A = 100o

Also given that AD âŠ¥ BC

Consider â–³ABD

We know that the sum of all three angles of a triangle is 180o.

Thus,

âˆ ABD + âˆ BAD + âˆ ADB = 180oÂ (Sum of angles of â–³ABD)

Or,

âˆ ABD + 50o + 90o = 180o

âˆ ABD =180o â€“Â 140o

âˆ ABD = 40o

21. In â–³ABC,Â âˆ A = 50o, âˆ B = 70oÂ and bisector of âˆ CÂ meets AB in D. Find the angles of the triangles ADC and BDC

Solution:

We know that the sum of all three angles of a triangle is equal to 180o.

Therefore, for the given â–³ABC, we can say that:

âˆ A + âˆ B + âˆ C = 180o (Sum of angles of â–³ABC)

50o + 70o + âˆ CÂ = 180o

âˆ C= 180o â€“120o

âˆ C = 60o

âˆ ACD = âˆ BCD =âˆ C2 (CD bisects âˆ C and meets AB in D.)

âˆ ACDÂ = âˆ BCDÂ = 60/2= 30o

Using the same logic for the given â–³ACD, we can say that:

âˆ DACÂ + âˆ ACDÂ + âˆ ADCÂ = 180o

50o + 30o + âˆ ADC = 180o

If we use the same logic for the given â–³BCD, we can say that

âˆ DBCÂ + âˆ BCDÂ + âˆ BDC = 180o

70o + 30o + âˆ BDCÂ = 180o

âˆ BDCÂ = 180o â€“ 100o

âˆ BDCÂ = 80o

Thus,

For â–³ADC: âˆ A = 50o, âˆ D = 100oÂ âˆ C = 30o

â–³BDC: âˆ B = 70o, âˆ D = 80oÂ âˆ C = 30o

22. In âˆ†ABC, âˆ A = 60o, âˆ B = 80o, and the bisectors of âˆ BÂ and âˆ C, meet at O. Find

(i) âˆ C

(ii) âˆ BOC

Solution:

(i) We know that the sum of all three angles of a triangle is 180o.

Hence, for â–³ABC, we can say that:

âˆ A + âˆ B + âˆ C = 180o (Sum of angles of âˆ†ABC)

60o + 80o + âˆ C= 180o.

âˆ C = 180o â€“ 140o

âˆ C = 40o.

(ii)For â–³OBC,

âˆ OBC = âˆ B/2 = 80/2 (OB bisects âˆ B)

âˆ OBC = 40o

âˆ OCB =âˆ C/2 = 40/2 (OC bisects âˆ C)

âˆ OCB = 20o

If we apply the above logic to this triangle, we can say that:

âˆ OCB + âˆ OBC + âˆ BOC = 180o (Sum of angles of â–³OBC)

20o + 40o + âˆ BOC = 180o

âˆ BOC = 180o â€“ 60o

âˆ BOC = 120o

23. The bisectors of the acute angles of a right triangle meet at O. Find the angle at O between the two bisectors.

Solution:

Given bisectors of the acute angles of a right triangle meet at O

We know that the sum of all three angles of a triangle is 180o.

Hence, for â–³ABC, we can say that:

âˆ A + âˆ B + âˆ C = 180o

âˆ AÂ + 90oÂ + âˆ CÂ = 180o

âˆ AÂ + âˆ CÂ = 180o â€“ 90o

âˆ AÂ + âˆ CÂ = 90o

For â–³OAC:

âˆ OACÂ = âˆ A/2Â (OA bisects âˆ A)

âˆ OCAÂ Â  = âˆ C/2 (OC bisects âˆ C)

On applying the above logic to â–³OAC, we get

âˆ AOC + âˆ OAC + âˆ OCAÂ = 180o (Sum of angles of â–³AOC)

âˆ AOC + âˆ A/2 + âˆ C/2 = 180o

âˆ AOC + (âˆ A + âˆ C)/2 = 180o

âˆ AOC + 90/2 = 180o

âˆ AOC = 180o â€“ 45o

âˆ AOC = 135o

24. Â In â–³ABC, âˆ A = 50o and BC is produced to a point D. The bisectors of âˆ ABCÂ and âˆ ACD meet at E. Find âˆ E.

Solution:

In the given triangle,

âˆ ACD = âˆ A + âˆ B. (Exterior angle is equal to the sum of two opposite interior angles.)

We know that the sum of all three angles of a triangle is 180o.

Therefore, for the given triangle, we know that the sum of the angles = 180o

âˆ ABC + âˆ BCA + âˆ CAB = 180o

âˆ A + âˆ B + âˆ BCA = 180o

âˆ BCA = 180o â€“ (âˆ A + âˆ B)

But we know that EC bisects âˆ ACD

Therefore âˆ ECA = âˆ ACD/2

âˆ ECA = (âˆ A + âˆ B)/2 [âˆ ACD = (âˆ A + âˆ B)]

But EB bisects âˆ ABC

âˆ EBC = âˆ ABC/2 = âˆ B/2

âˆ EBC = âˆ ECA + âˆ BCA

âˆ EBC = (âˆ A + âˆ B)/2 + 180o – (âˆ A + âˆ B)

If we use same steps for â–³EBC, then we get,

âˆ B/2 + (âˆ A + âˆ B)/2 + 180o – (âˆ A + âˆ B) + âˆ BEC = 180o

âˆ BEC = âˆ A + âˆ B – (âˆ A + âˆ B)/2 – âˆ B/2

âˆ BEC = âˆ A/2

âˆ BEC = 50o/2

= 25o

25. In âˆ†ABC, âˆ B = 60Â°, âˆ C = 40Â°, ALÂ âŠ¥ BC and AD bisects âˆ AÂ such that L and D lie on side BC. Find âˆ LAD

Solution:

We know that the sum of all angles of a triangle is 180Â°

Consider â–³ABC, we can write as

âˆ A + âˆ B + âˆ C = 180o

âˆ A + 60o + 40o = 180o

âˆ A = 800

But we know that âˆ DAC bisects âˆ A

âˆ DAC = âˆ A/2

âˆ DAC = 80o/2

If we apply same steps for the â–³ADC, we get

We know that the sum of all angles of a triangle is 180Â°

âˆ ADC + âˆ DCA + âˆ DAC = 180o

âˆ ADC + 40o + 40o = 180o

âˆ ADC = 180o – 80o = 1000

We know that exterior angle is equal to the sum of two interior opposite angles

Therefore we have

But here AL perpendicular to BC

100o = 90o + âˆ LAD

26. Line segments AB and CD intersect at O such that AC âˆ¥ DB. It âˆ CAB = 35o and âˆ CDB = 55o. Find âˆ BOD.

Solution:

We know that AC parallel to BD and AB cuts AC and BD at A and B, respectively.

âˆ CAB = âˆ DBA (Alternate interior angles)

âˆ DBA = 35o

We also know that the sum of all three angles of a triangle is 180o.

Hence, for â–³OBD, we can say that:

âˆ DBO + âˆ ODB + âˆ BOD = 180Â°

35o + 55o + âˆ BOD = 180o (âˆ DBO = âˆ DBA and âˆ ODB = âˆ CDB)

âˆ BOD = 180o – 90o

âˆ BOD = 90o

27. In Fig. 22, âˆ†ABCÂ is right angled at A, Q and R are points on line BC and P is a point such that QP âˆ¥ AC and RP âˆ¥ AB. Find âˆ P

Solution:

In the given triangle, AC parallel to QP and BR cuts AC and QP at C and Q, respectively.

âˆ QCA = âˆ CQP (Alternate interior angles)

Because RP parallel to AB and BR cuts AB and RP at B and R, respectively,

âˆ ABC = âˆ PRQ (alternate interior angles).

We know that the sum of all three angles of a triangle is 180o.

Hence, for âˆ†ABC, we can say that:

âˆ ABC + âˆ ACB + âˆ BAC = 180o

âˆ ABC + âˆ ACB + 90o = 180o (Right angled at A)

âˆ ABC + âˆ ACB = 90o

Using the same logic for â–³PQR, we can say that:

âˆ PQR + âˆ PRQ + âˆ QPR = 180o

âˆ ABC + âˆ ACB + âˆ QPR = 180o (âˆ ACB = âˆ PQR and âˆ ABC = âˆ PRQ)

Or,

90o+ âˆ QPR =180o (âˆ ABC+ âˆ ACB = 90o)

âˆ QPR = 90o

Exercise 15.3 Page No: 15.19

1. In Fig. 35, âˆ CBX is an exterior angle of âˆ†ABC at B. Name

(ii) The interior opposite angles to exterior âˆ CBX

Also, name the interior opposite angles to an exterior angle at A.

Solution:

(i) The interior adjacent angle is âˆ ABC

(ii) The interior opposite angles to exterior âˆ CBX is âˆ BAC and âˆ ACB

Also the interior angles opposite to exterior are âˆ ABC and âˆ ACB

2. In the fig. 36, two of the angles are indicated. What are the measures of âˆ ACX and âˆ ACB?

Solution:

Given that in â–³ABC, âˆ A = 50o and âˆ B = 55o

We know that the sum of angles in a triangle is 180o

Therefore we have

âˆ A + âˆ B + âˆ C = 180o

50o+ 55o+ âˆ C = 180o

âˆ C = 75o

âˆ ACB = 75o

âˆ ACX = 180oâˆ’ âˆ ACB = 180o âˆ’ 75o = 105o

3. In a triangle, an exterior angle at a vertex is 95o and its one of the interior opposite angles is 55o. Find all the angles of the triangle.

Solution:

We know that the sum of interior opposite angles is equal to the exterior angle.

Hence, for the given triangle, we can say that:

âˆ ABC+ âˆ BAC = âˆ BCO

55o + âˆ BAC = 95o

âˆ BAC= 95oâ€“ 55o

âˆ BAC = 40o

We also know that the sum of all angles of a triangle is 180o.

Hence, for the given â–³ABC, we can say that:

âˆ ABC + âˆ BAC + âˆ BCA = 180o

55o + 40o + âˆ BCA = 180o

âˆ BCA = 180o â€“95o

âˆ BCA = 85o

4. One of the exterior angles of a triangle is 80o, and the interior opposite angles are equal to each other. What is the measure of each of these two angles?

Solution:

Let us assume that A and B are the two interior opposite angles.

We know that âˆ A is equal to âˆ B.

We also know that the sum of interior opposite angles is equal to the exterior angle.

Therefore from the figure we have,

âˆ A + âˆ B = 80o

âˆ A +âˆ A = 80o (because âˆ A = âˆ B)

2âˆ A = 80o

âˆ A = 80/2 =40o

âˆ A= âˆ B = 40o

Thus, each of the required angles is of 40o.

5. The exterior angles, obtained on producing the base of a triangle both ways are 104o and 136o. Find all the angles of the triangle.

Solution:

In the given figure, âˆ ABE and âˆ ABC form a linear pair.

âˆ ABE + âˆ ABC =180o

âˆ ABC = 180oâ€“ 136o

âˆ ABC = 44o

We can also see that âˆ ACD and âˆ ACB form a linear pair.

âˆ ACD + âˆ ACB = 180o

âˆ ACB = 180oâ€“ 104o

âˆ ACB = 76o

We know that the sum of interior opposite angles is equal to the exterior angle.

Therefore, we can write as

âˆ BAC + âˆ ABC = 104o

âˆ BAC = 104o â€“ 44o = 60o

Thus,

âˆ ACE = 76o and âˆ BAC = 60o

6. In Fig. 37, the sides BC, CA and BA of a â–³ABC have been produced to D, E and F respectively. If âˆ ACD = 105o and âˆ EAF = 45o; find all the angles of the â–³ABC.

Solution:

In a â–³ABC, âˆ BAC and âˆ EAF are vertically opposite angles.

Hence, we can write as

âˆ BAC = âˆ EAF = 45o

Considering the exterior angle property, we have

âˆ BAC + âˆ ABC = âˆ ACD = 105o

On rearranging we get

âˆ ABC = 105oâ€“ 45o = 60o

We know that the sum of angles in a triangle is 180o

âˆ ABC + âˆ ACB +âˆ BAC = 180Â°

âˆ ACB = 75o

Therefore, the angles are 45o, 60o and 75o.

7. In Fig. 38, AC perpendicular to CE and C âˆ A: âˆ B: âˆ C= 3: 2: 1. Find the value of âˆ ECD.

Solution:

In the given triangle, the angles are in the ratio 3: 2: 1.

Let the angles of the triangle be 3x, 2x and x.

We know that sum of angles in a triangle is 180o

3x + 2x + x = 180o

6x = 180o

x = 30o

Also, âˆ ACB + âˆ ACE + âˆ ECD = 180o

x + 90o + âˆ ECD = 180o (âˆ ACE = 90o)

We know that x = 30o

Therefore

âˆ ECD = 60o

8. A student when asked to measure two exterior angles of â–³ABC observed that the exterior angles at A and B are of 103o and 74o respectively. Is this possible? Why or why not?

Solution:

We know that sum of internal and external angle is equal to 180o

Internal angle at A + External angle at A = 180o

Internal angle at A + 103o =180o

Internal angle at A = 77o

Internal angle at B + External angle at B = 180o

Internal angle at B + 74o = 180o

Internal angle at B = 106o

Sum of internal angles at A and B = 77o + 106o = 183o

It means that the sum of internal angles at A and B is greater than 180o, which cannot be possible.

9. In Fig.39, AD and CF are respectively perpendiculars to sides BC and AB of â–³ABC. If âˆ FCD = 50o, find âˆ BAD

Solution:

We know that the sum of all angles of a triangle is 180o

Therefore, for the given â–³FCB, we have

âˆ FCB + âˆ CBF + âˆ BFC = 180o

50o + âˆ CBF + 90o = 180o

âˆ CBF = 180o â€“ 50o â€“ 90o = 40o

Using the above steps for â–³ABD, we can say that:

âˆ ABD + âˆ BDA + âˆ BAD = 180o

âˆ BAD = 180o â€“ 90o â€“ 40o = 50o

10. In Fig.40, measures of some angles are indicated. Find the value of x.

Solution:

We know that the sum of the angles of a triangle is 180o

From the figure we have,

âˆ AED + 120o = 180o (Linear pair)

âˆ AED = 180o â€“ 120o = 60o

We know that the sum of all angles of a triangle is 180o.

âˆ ADE + âˆ AED + âˆ DAE = 180o

60o+ âˆ ADE + 30o =180o

âˆ ADE = 180o â€“ 60o â€“ 30o = 90o

From the given figure, we have

âˆ FDC + 90o = 180o (Linear pair)

âˆ FDC = 180o â€“ 90o = 90o

Using the same steps for â–³CDF, we get

âˆ CDF + âˆ DCF + âˆ DFC = 180o

90o + âˆ DCF + 60o = 180o

âˆ DCF = 180o â€“ 60o â€“ 90o = 30o

Again from the figure we have

âˆ DCF + x = 180o (Linear pair)

30o + x = 180o

x = 180o â€“ 30o = 150o

11. In Fig. 41, ABC is a right triangle right angled at A. D lies on BA produced and DE perpendicular to BC intersecting AC at F. If âˆ AFE = 130o, find

(i) âˆ BDE

(ii) âˆ BCA

(iii) âˆ ABC

Solution:

(i) Here,

âˆ BAF + âˆ FAD = 180o (Linear pair)

âˆ FAD = 180o – âˆ BAF = 180o â€“ 90o = 90o

Also from the figure,

âˆ ADF + 90o = 130o

âˆ ADF = 130o âˆ’ 90o = 40o

âˆ BDE = 40o

(ii) We know that the sum of all the angles of a triangle is 180o.

Therefore, for â–³BDE, we have

âˆ BDE + âˆ BED + âˆ DBE = 180o

âˆ DBE = 180o â€“ âˆ BDE – âˆ BED

âˆ DBE = 180o â€“ 40o â€“ 90o = 50o â€¦. Equation (i)

Again from the figure we have,

âˆ FAD = âˆ ABC + âˆ ACB (Exterior angle property)

90o = 50o + âˆ ACB

âˆ ACB = 90o â€“ 50o = 40o

(iii) From equation we have

âˆ ABC = âˆ DBE = 50o

12. ABC is a triangle in which âˆ B = âˆ C and ray AX bisects the exterior angle DAC. If âˆ DAX = 70o. Find âˆ ACB.

Solution:

Given that ABC is a triangle in which âˆ B = âˆ C

Also given that AX bisects the exterior angle DAC

âˆ CAX = âˆ DAX (AX bisects âˆ CAD)

âˆ CAX =70o [given]

âˆ CAX +âˆ DAX + âˆ CAB =180o

70o+ 70o + âˆ CAB =180o

âˆ CAB =180o â€“140o

âˆ CAB =40o

âˆ ACB + âˆ CBA + âˆ CAB = 180o (Sum of the angles of âˆ†ABC)

âˆ ACB + âˆ ACB+ 40o = 180o (âˆ C = âˆ B)

2âˆ ACB = 180o â€“ 40o

âˆ ACB = 140/2

âˆ ACB = 70o

13. The side BC of â–³ABC is produced to a point D. The bisector of âˆ A meets side BC in L. If âˆ ABC= 30o and âˆ ACD = 115o, find âˆ ALC

Solution:

Given that âˆ ABC= 30o and âˆ ACD = 115o

From the figure, we have

âˆ ACD and âˆ ACL make a linear pair.

âˆ ACD+ âˆ ACB = 180o

115o + âˆ ACB =180o

âˆ ACB = 180o â€“ 115o

âˆ ACB = 65o

We know that the sum of all angles of a triangle is 180o.

Therefore, forâ–³ ABC, we have

âˆ ABC + âˆ BAC + âˆ ACB = 180o

30o + âˆ BAC + 65o = 180o

âˆ BAC = 85o

âˆ LAC = âˆ BAC/2 = 85/2

Using the same steps for â–³ALC, we get

âˆ ALC + âˆ LAC + âˆ ACL = 180o

âˆ ALC + 85/2 + 65o = 180o

We know that âˆ ACL = âˆ ACB

âˆ ALC = 180o – 85/2 – 65o

âˆ ALC = 72 Â½o

14. D is a point on the side BC of â–³ABC. A line PDQ through D, meets side AC in P and AB produced at Q. If âˆ A = 80o, âˆ ABC = 60o and âˆ PDC = 15o, find

(i) âˆ AQD

(ii) âˆ APD

Solution:

From the figure we have

âˆ ABD and âˆ QBD form a linear pair.

âˆ ABC + âˆ QBC =180o

60o + âˆ QBC = 180o

âˆ QBC = 120o

âˆ PDC = âˆ BDQ (Vertically opposite angles)

âˆ BDQ = 15o

(i) In â–³QBD:

âˆ QBD + âˆ QDB + âˆ BQD = 180o (Sum of angles of â–³QBD)

120o+ 15o + âˆ BQD = 180o

âˆ BQD = 180o â€“ 135o

âˆ BQD = 45o

âˆ AQD = âˆ BQD = 45o

(ii) In â–³AQP:

âˆ QAP + âˆ AQP + âˆ APQ = 180o (Sum of angles of â–³AQP)

80o + 45o + âˆ APQ = 180o

âˆ APQ = 55o

âˆ APD = âˆ APQ = 55o

15. Explain the concept of interior and exterior angles and in each of the figures given below. Find x and y (Fig. 42)

Solution:

The interior angles of a triangle are the three angle elements inside the triangle.

The exterior angles are formed by extending the sides of a triangle, and if the side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

Using these definitions, we will obtain the values of x and y.

(i) From the given figure, we have

âˆ ACB + x = 180o (Linear pair)

75o+ x = 180o

x = 105o

We know that the sum of all angles of a triangle is 180o

Therefore, for â–³ABC, we can say that:

âˆ BAC+ âˆ ABC +âˆ ACB = 180o

40o+ y +75o = 180o

y = 65o

(ii) From the figure, we have

x + 80o = 180o (Linear pair)

x = 100o

In â–³ABC, we have

We also know that the sum of angles of a triangle is 180o

x + y + 30o = 180o

100o + 30o + y = 180o

y = 50o

(iii) We know that the sum of all angles of a triangle is 180Â°.

Therefore, for â–³ACD, we have

30o + 100o + y = 180o

y = 50o

Again from the figure we can write as

âˆ ACB + 100Â° = 180Â°

âˆ ACB = 80o

Using the above rule for â–³ACB, we can say that:

x + 45o + 80o = 180o

x = 55o

(iv) We know that the sum of all angles of a triangle is 180o.

Therefore, for â–³DBC, we have

30o + 50o + âˆ DBC = 180o

âˆ DBC = 100o

From the figure we can say that

x + âˆ DBC = 180o is a Linear pair

x = 80o

From the exterior angle property we have

y = 30o + 80o = 110o

16. Compute the value of x in each of the following figures:

Solution:

(i) From the given figure, we can write as

âˆ ACD + âˆ ACB = 180o is a linear pair

On rearranging we get

âˆ ACB = 180o â€“ 112o = 68o

Again from the figure we have,

âˆ BAE + âˆ BAC = 180o is a linear pair

On rearranging we get,

âˆ BAC = 180oâ€“ 120o = 60o

We know that the sum of all angles of a triangle is 180o.

Therefore, for â–³ABC:

x + âˆ BAC + âˆ ACB = 180o

x = 180o â€“ 60o â€“ 68o = 52o

x = 52o

(ii) From the given figure, we can write as

âˆ ABC + 120o = 180o is a linear pair

âˆ ABC = 60o

Again from the figure we can write as

âˆ ACB+ 110o = 180o is a linear pair

âˆ ACB = 70o

We know that the sum of all angles of a triangle is 180o.

Therefore, consider â–³ABC, we get

x + âˆ ABC + âˆ ACB = 180o

x = 50o

(iii) From the given figure, we can write as

We know that the sum of all the angles of a triangle is 180o.

Therefore, consider â–³DEC, we have

x + 40o + 52o = 180o

x = 88o

(iv) In the given figure, we have a quadrilateral and also we know that sum of all angles in a quadrilateral is 360o.

Thus,

35o + 45o + 50o + reflex âˆ ADC = 360o

On rearranging we get,

230o + x = 360o (A complete angle)

x = 130o

Exercise 15.4 Page No: 15.24

1. In each of the following, there are three positive numbers. State if these numbers could possibly be the lengths of the sides of a triangle:

(i) 5, 7, 9

(ii) 2, 10, 15

(iii) 3, 4, 5

(iv) 2, 5, 7

(v) 5, 8, 20

Solution:

(i) Given 5, 7, 9

Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side.

Here, 5 + 7 > 9, 5 + 9 > 7, 9 + 7 > 5

(ii) Given 2, 10, 15

No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.

Here, 2 + 10 < 15

(iii) Given 3, 4, 5

Yes, these numbers can be the lengths of the sides of a triangle because the sum of any two sides of triangle is always greater than the third side.

Here, 3 + 4 > 5, 3 + 5 > 4, 4 + 5 > 3

(iv) Given 2, 5, 7

No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.

Here, 2 + 5 = 7

(v) Given 5, 8, 20

No, these numbers cannot be the lengths of the sides of a triangle because the sum of any two sides of a triangle is always greater than the third side, which is not true in this case.

Here, 5 + 8 < 20

2. In Fig. 46, P is the point on the side BC. Complete each of the following statements using symbol â€˜=â€™,â€™ > â€˜or â€˜< â€˜so as to make it true:

(i) APâ€¦ AB+ BP

(ii) APâ€¦ AC + PC

(iii) APâ€¦. Â½ (AB + AC + BC)

Solution:

(i) In â–³APB, AP < AB + BP because the sum of any two sides of a triangle is greater than the third side.

(ii) In â–³APC, AP < AC + PC because the sum of any two sides of a triangle is greater than the third side.

(iii) AP < Â½ (AB + AC + BC)

In â–³ABP and â–³ACP, we can write as

AP < AB + BPâ€¦ (i) (Because the sum of any two sides of a triangle is greater than the third side)

AP < AC + PC â€¦ (ii) (Because the sum of any two sides of a triangle is greater than the third side)

On adding (i) and (ii), we have:

AP + AP < AB + BP + AC + PC

2AP < AB + AC + BC (BC = BP + PC)

AP < Â½ (AB + AC + BC)

3. P is a point in the interior of â–³ABC as shown in Fig. 47. State which of the following statements are true (T) or false (F):

(i) AP + PB < AB

(ii) AP + PC > AC

(iii) BP + PC = BC

Solution:

(i) False

Explanation:

We know that the sum of any two sides of a triangle is greater than the third side, it is not true for the given triangle.

(ii) True

Explanation:

We know that the sum of any two sides of a triangle is greater than the third side, it is true for the given triangle.

(iii) False

Explanation:

We know that the sum of any two sides of a triangle is greater than the third side, it is not true for the given triangle.

4. O is a point in the exterior of â–³ABC. What symbol â€˜>â€™,â€™<â€™ or â€˜=â€™ will you see to complete the statement OA+OBâ€¦.AB? Write two other similar statements and show that OA + OB + OC > Â½ (AB + BC +CA)

Solution:

We know that the sum of any two sides of a triangle is always greater than the third side, in â–³OAB, we have,

OA + OB > AB â€¦.. (i)

In â–³OBC we have

OB + OC > BC â€¦â€¦ (ii)

In â–³OCA we have

OA + OC > CA â€¦.. (iii)

On adding equations (i), (ii) and (iii) we get:

OA + OB + OB + OC + OA + OC > AB + BC + CA

2(OA + OB + OC) > AB + BC + CA

OA + OB + OC > (AB + BC + CA)/2

Or

OA + OB + OC > Â½ (AB + BC +CA)

Hence the proof.

5. InÂ â–³ABC, âˆ A = 100o, âˆ B = 30o, âˆ C = 50o. Name the smallest and the largest sides of the triangle.

Solution:

We know that the smallest side is always opposite to the smallest angle, which in this case is 30Â°, it is AC.

Also, because the largest side is always opposite to the largest angle, which in this case is 100Â°, it is BC.

Exercise 15.5 Page No: 15.30

1. State Pythagoras theorem and its converse.

Solution:

The Pythagoras Theorem:

In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the other two sides.

Converse of the Pythagoras Theorem:

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle, with the angle opposite to the first side as right angle.

2. In rightÂ â–³ABC, the lengths of the legs are given. Find the length of the hypotenuse

(i) a = 6 cm, b = 8 cm

(ii) a = 8 cm, b = 15 cm

(iii) a = 3 cm, b = 4 cm

(iv) a = 2 cm, b =1.5 cm

Solution:

(i) According to the Pythagoras theorem, we have

(Hypotenuse)2Â = (Base)2Â + (Height)2

Let c be hypotenuse and a and b be other two legs of right angled triangle

Then we have

c2Â = a2Â + b2

c2Â = 62Â + 82

c2Â = 36 + 64 = 100

c = 10 cm

(ii) According to the Pythagoras theorem, we have

(Hypotenuse)2Â = (Base)2Â + (Height)2

Let c be hypotenuse and a and b be other two legs of right angled triangle

Then we have

c2Â = a2Â + b2

c2Â = 82Â + 152

c2Â = 64 + 225 = 289

c = 17cm

(iii) According to the Pythagoras theorem, we have

(Hypotenuse)2Â = (Base)2Â + (Height)2

Let c be hypotenuse and a and b be other two legs of right angled triangle

Then we have

c2Â = a2Â + b2

c2Â = 32Â + 42

c2Â = 9 + 16 = 25

c = 5 cm

(iv) According to the Pythagoras theorem, we have

(Hypotenuse)2Â = (Base)2Â + (Height)2

Let c be hypotenuse and a and b be other two legs of right angled triangle

Then we have

c2Â = a2Â + b2

c2Â = 22Â + 1.52

c2Â = 4 + 2.25 = 6.25

c = 2.5 cm

3. The hypotenuse of a triangle is 2.5 cm. If one of the sides is 1.5 cm. find the length of the other side.

Solution:

Let c be hypotenuse and the other two sides be b and a

According to the Pythagoras theorem, we have

c2Â = a2Â + b2

2.52 = 1.52 + b2

b2Â = 6.25 âˆ’2.25 = 4

b = 2 cm

Hence, the length of the other side is 2 cm.

4. A ladder 3.7 m long is placed against a wall in such a way that the foot of the ladder is 1.2 m away from the wall. Find the height of the wall to which the ladder reaches.

Solution:

Let the height of the ladder reaches to the wall be h.

According to the Pythagoras theorem, we have

(Hypotenuse)2Â = (Base)2Â + (Height)2

3.72 = 1.22 + h2

h2Â = 13.69 â€“ 1.44 = 12.25

h = 3.5 m

Hence, the height of the wall is 3.5 m.

5. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is right-angled triangle.

Solution:

In the given triangle, the largest side is 6 cm.

We know that in a right angled triangle, the sum of the squares of the smaller sides should be equal to the square of the largest side.

Therefore,

32Â + 42 = 9 + 16 = 25

But, 62Â = 36

32Â + 42Â = 25 which is not equal to 62

Hence, the given triangle is not a right angled triangle.

6. The sides of certain triangles are given below. Determine which of them are right triangles.

(i) a = 7 cm, b = 24 cm and c= 25 cm

(ii) a = 9 cm, b = 16 cm and c = 18 cm

Solution:

(i) We know that in a right angled triangle, the square of the largest side is equal to the sum of the squares of the smaller sides.

Here, the larger side is c, which is 25 cm.

c2Â = 625

Given that,

a2+ b2Â = 72Â + 242

= 49 + 576

= 625

= c2

Thus, the given triangle is a right triangle.

(ii) We know that in a right angled triangle, the square of the largest side is equal to the sum of the squares of the smaller sides.

Here, the larger side is c, which is 18 cm.

c2Â = 324

Given that

a2+ b2Â = 92 + 162Â

= 81 + 256

= 337 which is not equal to c2

Thus, the given triangle is not a right triangle.

7. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m.Â  Find the distance between their tops.

(Hint: Find the hypotenuse of a right triangle having the sides (11 â€“ 6) m = 5 m and 12 m)

Solution:

Let the distance between the tops of the poles is the distance between points A and B.

We can see from the given figure that points A, B and C form a right triangle, with AB as the hypotenuse.

By using the Pythagoras Theorem in â–³ABC, we get

(11âˆ’6)2Â + 122Â = AB2

AB2Â = 25 + 144

AB2Â = 169

AB = 13

Hence, the distance between the tops of the poles is 13 m.

8. A man goes 15 m due west and then 8 m due north. How far is he from the starting point?

Solution:

Given a man goes 15 m due west and then 8 m due north

Let O be the starting point and P be the final point.

Then OP becomes the hypotenuse in the triangle.

So by using the Pythagoras theorem, we can find the distance OP.

OP2Â = 152Â + 82

OP2Â = 225 + 64

OP2Â = 289

OP = 17

Hence, the required distance is 17 m.

9. The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reach?

Solution:

Given Let the length of the ladder be L m.

By using the Pythagoras theorem, we can find the length of the ladder.

62Â + 82Â = L2

L2Â = 36 + 64 =100

L = 10

Thus, the length of the ladder is 10 m.

Let the height of the ladder after it is shifted be H m.

By using the Pythagoras theorem, we can find the height of the ladder after it is shifted.

82Â + H2Â = 102

H2Â = 100 â€“ 64 = 36

H = 6

Thus, the height of the ladder is 6 m.

10. A ladder 50 dm long when set against the wall of a house just reaches a window at a height of 48 dm. How far is the lower end of the ladder from the base of the wall?

Solution:

Given that length of a ladder is 50dm

Let the distance of the lower end of the ladder from the wall be x dm.

By using the Pythagoras theorem, we get

x2Â + 482Â = 502

x2Â = 502Â âˆ’ 482Â

= 2500 â€“ 2304

= 196

H = 14 dm

Hence, the distance of the lower end of the ladder from the wall is 14 dm.

11. The two legs of a right triangle are equal and the square of the hypotenuse is 50. Find the length of each leg.Â

Solution:

According to the Pythagoras theorem, we have

(Hypotenuse)2Â = (Base)2Â + (Height)2

Given that the two legs of a right triangle are equal and the square of the hypotenuse, which is 50

Let the length of each leg of the given triangle be x units.

Using the Pythagoras theorem, we get

x2Â + x2Â = (Hypotenuse)2

x2Â + x2Â = 50

2x2Â = 50

x2Â = 25

x = 5

Hence, the length of each leg is 5 units.

12. Verity that the following numbers represent Pythagorean triplet:

(i) 12, 35, 37

(ii) 7, 24, 25

(iii) 27, 36, 45

(iv) 15, 36, 39

Solution:

(i) The condition for Pythagorean triplet is the square of the largest side is equal to the sum of the squares of the other two sides.

372 =1369

122 + 352 = 144 + 1225 = 1369

122 + 352 = 372

Yes, they represent a Pythagorean triplet.

(ii) The condition for Pythagorean triplet is the square of the largest side is equal to the sum of the squares of the other two sides.

252Â = 625

72Â + 242Â = 49 + 576 = 625

72Â + 242Â = 252

Yes, they represent a Pythagorean triplet.

(iii) The condition for Pythagorean triplet is the square of the largest side is equal to the sum of the squares of the other two sides.

452Â = 2025

272Â + 362Â = 729 + 1296 = 2025

272Â + 362Â = 452

Yes, they represent a Pythagorean triplet.

(iv) The condition for Pythagorean triplet is the square of the largest side is equal to the sum of the squares of the other two sides.

392Â = 1521

152Â + 362Â = 225 + 1296 = 1521

152Â + 362Â = 392

Yes, they represent a Pythagorean triplet.

13. In â–³ABC, âˆ ABC = 100Â°, âˆ BAC = 35Â°Â and BD âŠ¥ AC meets side AC in D. If BD = 2 cm, find âˆ C, and length DC.

Solution:

We know that the sum of all angles of a triangle is 180o

Therefore, for the given â–³ABC, we can say that:

âˆ ABC + âˆ BAC + âˆ ACB = 180o

100o + 35o + âˆ ACB = 180o

âˆ ACB = 180o â€“135o

âˆ ACB = 45o

âˆ C = 45o

On applying same steps for the â–³BCD, we get

âˆ BCD + âˆ BDC + âˆ CBD = 180o

45o + 90o + âˆ CBD = 180o (âˆ ACB = âˆ BCD and BD is perpendicular to AC)

âˆ CBD = 180o â€“ 135o

âˆ CBD = 45o

We know that the sides opposite to equal angles have equal length.

Thus, BD = DC

DC = 2 cm

14. In a â–³ABC, AD is the altitude from A such that AD = 12 cm. BD = 9 cm and DC = 16 cm. Examine if â–³ABC is right angled at A.

Solution:

Using the Pythagoras theorem, we get

122Â + 162Â = AC2

AC2Â = 144 + 256

= 400

AC = 20 cm

Using the Pythagoras theorem, we get

122Â + 92Â = AB2

AB2Â = 144 + 81 = 225

AB = 15 cm

Consider â–³ABC,

BC2Â = 252Â = 625

AB2Â + AC2Â = 152Â + 202Â = 625

AB2Â + AC2Â = BC2

Because it satisfies the Pythagoras theorem, therefore â–³ABC is right angled at A.

15. Draw a triangle ABC, with AC = 4 cm, BC = 3 cm and âˆ C = 105o. Measure AB. Is (AB)2Â = (AC)2Â  + (BC)2? If not which one of the following is true:

(AB)2Â > (AC)2Â + (BC)2Â or (AB)2Â < (AC)2Â + (BC)2?

Solution:

Draw â–³ABC as shown in the figure with following steps.

Draw a line BC = 3 cm.

At point C, draw a line at 105o angle with BC.

Take an arc of 4 cm from point C, which will cut the line at point A.

Now, join AB, which will be approximately 5.5 cm.

AC2Â + BC2Â = 42Â + 32Â

= 9 + 16

= 25

AB2Â = 5.52 = 30.25

AB2Â is not equal to AC2+ BC2

Therefore we have

AB2Â > AC2Â + BC2

16. Draw a triangle ABC, with AC = 4 cm, BC = 3 cm and âˆ C = 80Â°. Measure AB. Is (AB)2Â = (AC)2Â  + (BC)2? If not which one of the following is true:

(AB)2Â > (AC)2Â + (BC)2Â or (AB)2Â < (AC)2Â + (BC)2?

Solution:

Draw â–³ABC as shown in the figure with following steps.

Draw a line BC = 3 cm.

At point C, draw a line at 80Â° angle with BC.

Take an arc of 4 cm from point C, which will cut the line at point A.

Now, join AB, it will be approximately 4.5 cm.

AC2Â + BC2Â = 42Â + 32

= 9 +16

= 25

AB2Â = (4.5)2

= 20.25

AB2Â not equal to AC2Â + BC2

Therefore here AB2Â < AC2Â + BC2