Question 1:The areas of two similar triangles are 81 cm^{2} and 49 cm^{2} respectively. If the altitude of the first triangle is 6.3 cm, find the corresponding altitude of the other.

Solution:

Given: \(\Delta\)

ar(\(\Delta\)^{2} and ar(\(\Delta\)^{2}

Let AL and DM be the corresponding altitudes of \(\Delta\)

We know that the ratio of the area of two similar triangles is equal to the ratio of the square of corresponding altitudes:

\(\frac{ar(\Delta ABC)}{ar(\Delta DEF)}=\frac{AL^{2}}{DM^{2}}\)

\(\Rightarrow\)

\(\Rightarrow\)^{2} = \(\left ( \frac{49\times 6.3\times 6.3}{81} \right )=24.01\)

\(\Rightarrow\)

Hence, DM = 4.9 cm

Therefore, the required altitude is 4.9 cm

Question 2:The areas of two similar triangles are 100 cm^{2} and 64 cm^{2} respectively. If a median of the smaller triangle is 5.6 cm, find the corresponding median of the other.

Solution:

Given: \(\Delta\)^{2} and ar(\(\Delta\)^{2}

Let AP and DQ be the corresponding medians of \(\Delta\)

Let AP = x cm.

We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding medians.

Therefore, \(\frac{area of \Delta ABC}{area of \Delta DEF}=\frac{AP^{2}}{DQ^{2}}\)

\(\Rightarrow\)

\(\Rightarrow\)^{2} = \(\frac{100\times (5.6)^{2}}{64}\)

\(\Rightarrow\)

\(\Rightarrow\)

Hence, AP = 7 cm

Question 3:In the given figure, ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm, prove that area of \(\Delta\)

Solution:

Given: AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm

AB = AP + PB = (1 + 3) cm = 4 cm

AC = AQ + QC = (1.5 + 4.5) cm = 6 cm

Now, we have:

\(\frac{AP}{AB}=\frac{1}{4}\)

In \(\Delta\)

\(\angle\)

\(\angle\)

\(\Delta\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Hence proved.

Question 4:In the given figure, DE || BC. If DE = 3 cm, BC = 6 cm and ar(\(\Delta\)^{2}, find the area of \(\Delta\)

Solution:

Given DE || BC

DE = 3 cm and BC = 6 cm

ar(\(\Delta\)^{2}

In \(\Delta\)

\(\angle\)

\(\angle\)

Therefore, \(\Delta\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Therefore, \(ar(\Delta ABC)=60cm^{2}\)

Question 5:\(\Delta\)

Solution:

In \(\Delta\)^{o} (AD \(\perp\)

\(\angle\)

\(\Delta\)

\(\frac{ar(\Delta ABC)}{ar(\Delta ADC)}=\frac{ar(\Delta BAC)}{ar(\Delta ADC)}=\frac{BC^{2}}{AC^{2}}\)

\(\Rightarrow\)

Therefore, the ratio of the areas of \(\Delta\)

Question 6:In the given figure, DE || BC and DE : BC = 3 : 5. Calculate the ratio of the areas of \(\Delta\)

Solution:

Let DE = 3x and BC = 5x

In \(\Delta\)

\(\angle\)

\(\angle\)

\(\Delta\)

\(\Rightarrow\)

Let, ar(\(\Delta\)^{2} units

Then, ar(\(\Delta\)^{2} units

Therefore, \(ar(trap.BCED)=ar(\Delta ABC)-ar(\Delta ADE)=(25x^{2}-9x^{2})=16x^{2}units\)

Therefore, \(\frac{ar(\Delta ADE)}{ar(trap.BCED)}=\frac{9x^{2}}{16x^{2}}=\frac{9}{16}\)

Therefore, ratio of ar(\(\Delta\)

Question 7:In \(\Delta\)

Solution:

In \(\Delta\)

So, DE || BC and DE = ½ BC

Now, in \(\Delta\)

\(\angle\)

\(\angle\)

\(\Delta\)

Let AD = x and AB = 2x

Therefore, \(\frac{ar(\Delta ADE)}{ar(\Delta ABC)}=\frac{AD^{2}}{AB^{2}}\)

\(\Rightarrow\)

Therefore, the ratio of the areas of \(\Delta\)

## EXERCISE 4D

Question 1:The sides of certain triangles are given below. Determine which of them are right angles.

(i) 9 cm, 16 cm, 18 cm

(ii) 7 cm, 24 cm, 25 cm

(iii) 1.4 cm, 4.8 cm, 5 cm

(iv) 1.6 cm, 3.8 cm, 4 cm

(v)(a – 1) cm, \(2\sqrt{a}\)

Solution:

For a given triangle to be a right angled, the sum of the squares of the two sides must be equal to the square of the largest side.

(i)Let a = 9 cm, b = 16 cm and c = 18 cm. Then

(a^{2} + b^{2}) = [9^{2} + (16)^{2}] = (81 + 256) cm^{2} = 337 cm^{2}

And c^{2} = (18)^{2} cm^{2} = 324 cm^{2}

Therefore, (a^{2} + b^{2}) \(\neq\)^{2}

Hence, the given triangle is not right angled.

(ii)Let a = 7 cm, b = 24 cm and c = 25 cm, Then

(a^{2} + b^{2}) = [7^{2} + (24)^{2}] = (49 + 576) cm^{2} = 625 cm^{2}

And c^{2} = (25)^{2} cm^{2} = 625 cm^{2}

Therefore, (a^{2} + b^{2}) = c^{2}

Hence, the given triangle is a right triangle.

(iii)Let a = 1.4 cm, b = 4.8 cm, and c = 5 cm

(a^{2} + b^{2}) = [(1.4)^{2} + (4.8)^{2}] = (1.96 + 23.04) cm^{2} = 25 cm^{2}

And c^{2} = (5)^{2} cm^{2} = 25 cm^{2}

Therefore, (a^{2} + b^{2}) = c^{2}

Hence, the given triangle is a right triangle.

(iv)Let a = 1.6 cm, b = 3.8 cm and c = 4 cm

(a^{2} + b^{2}) = [(1.6)^{2} + (3.8)^{2}] = (2.56 + 14.44) cm^{2} = 17 cm^{2}

And c^{2} = (4)^{2} cm^{2} = 16 cm^{2}

Therefore, (a^{2} + b^{2}) \(\neq\)^{2}

Hence, the given triangle is not a right triangle.

(v)Let p = (a – 1) cm, q = \(2\sqrt{a}\)

(p^{2} + q^{2}) = [(a – 1)^{2} + (\(2\sqrt{a}\)^{2}] = (a^{2} + 1 – 2a + 4a) cm^{2} = (a + 1)^{2} cm^{2}

And r^{2} = (a + 1)^{2} cm^{2}

Therefore, (p^{2} + q^{2}) = r^{2}

Hence, the given triangle is a right triangle.

Question 2:A man goes 80 m due east and then 150 m due north. How far is he from the starting point?

Solution:

Starting from A, let the man goes from A to B and from B to C, as shown in the figure.

Then,

AB = 80 m, BC = 150 m and \(\angle\)^{o}

From right \(\Delta\)

By Pythagoras theorm, we have:

AC^{2} = (AB^{2} + BC^{2}) m^{2}

= [(80)^{2} + (150)^{2}] m^{2} = (6400 + 22500) m^{2} = 28900 m^{2}

Therefore, AC = \(\sqrt{28900}\)

Hence, the man is 170 m north-east from the starting point.

Question 3:A man goes 10 m due south and then 24 m due west. How far is he from the starting point?

Solution:

Starting from O, let the man goes from O to A and then A to B as shown in the figure.

Then,

OA = 10 m, AB = 24 m and \(\angle\)^{o}

Using Pythagoras theorem:

OB^{2} = OA^{2} + AB^{2}

\(\Rightarrow\)^{2} = 10^{2} + 24^{2}

\(\Rightarrow\)^{2} = 100 + 576

\(\Rightarrow\)^{2} = 676

\(\Rightarrow\)

Hence, the man is 26 m south-west from the starting position.

Question 4:A 13-m-long ladder reaches a window of a building 12 m above the ground. Determine the distance of the foot of the ladder from the building.

Solution:

Let AB be the building and CB be the ladder.

Then,

AB = 12 m, CB = 13 m and \(\angle\)^{o}

By Pythagoras theorem, we have:

CB^{2} = AB^{2} + AC^{2}

AC^{2} = [CB^{2} – AB^{2}] = [(13)^{2} – (12)^{2}] m^{2} = (169 – 144) m^{2} = 25 m^{2}

\(\Rightarrow\)

Hence, the distance of the foot of the ladder from the building is 5 m.

Question 5:A ladder is placed in such a way that its foot is at a distance of 15 m from a wall and its top reaches a window 20 m above the ground. Find the length of the ladder.

Solution:

Let AB be the wall where window is at B, CB be the ladder and AC be the distance between the foot of the ladder and wall.

Then,

AB = 20 m, AC = 15 m, and \(\angle\)^{o}

By Pythagoras theorem, we have

CB^{2} = AB^{2} + AC^{2} = [(20)^{2} + (15)^{2}] m^{2} = (400 + 225) m^{2} = 625 m^{2}

CB = \(\sqrt{625}\)

Hence, the length of ladder is 25 m.

Question 6: Two vertical poles of height 9 m and 14 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

Solution:

Let AB and CD be the given vertical poles.

Then,

AB = 9 m, CD = 14 m and AC = 12 m

Const:Draw, BE || AC.

Then,

CE = AB = 9 m and BE = AC = 12 m

DE = (CD – CE) = (14 – 9) = 5 m

In right \(\Delta\)

BD^{2} = BE^{2} + DE^{2} = [(12)^{2} + (5)^{2}] m^{2} = (144 + 25) m^{2} = 169 m^{2}

\(\Rightarrow\)

Hence, the distance between their tops is 13 m.

Question 7:In the given figure, O is a point inside a \(\Delta\)^{o}, OP = 6 cm and OR = 8 cm. If PQ = 24 cm and QR = 26 cm, prove that \(\Delta\)

Solution:

In \(\Delta\)^{o}, PQ = 24 cm, and QR = 26 cm^{2}

In \(\Delta\)^{o}

In \(\Delta\)

PR^{2} = PO^{2} + OR^{2}

PR^{2} = (6^{2} + 8^{2}) cm^{2} = (36 + 64) cm^{2} = 100 cm^{2}

PR = \(\sqrt{100}\)

In \(\Delta\)

By Pythagoras theorem, we have

QR^{2} = QP^{2} + PR^{2}

(26)^{2} cm^{2} = (24^{2} + 10^{2}) cm^{2}

676 cm^{2} = (576 + 100) cm^{2}

676 cm^{2 }= 676 cm^{2}

Hence, QR^{2} = QP^{2} + PR^{2}

(Sum of square of two sides equal to square of the greatest side)

Hence, \(\Delta\)

Question 8:\(\Delta\)

Solution:

In an equilateral triangle all sides are equal.

Then, AB = BC = AC = 2a units

Const: Draw an altitude AD \(\perp\)

Given BC = 2a. Then, BD = a

In \(\Delta\)

\(\angle\)^{o}

(AB)^{2} = (AD)^{2} + (BD)^{2} [by Pythagoras thorem]

(AD)^{2} = (AB^{2} – BD^{2})

= [(2a)^{2} – (a)^{2}] sq. unit = (4a^{2} – a^{2}) sq. unit

AD = \(\sqrt{3a^{2}}\)

Question 9:Find the height of an equilateral triangle of side 12 cm.

Solution:

\(\Delta\)

Therefore, AB = BC = AC = 12 cm

If BC = 12 cm

Then, BD = DC = 6 cm

In \(\Delta\)

AB^{2} = AD^{2} + BD^{2}

AD^{2} = [(12)^{2} – (6]^{2}] cm^{2}

AD^{2} = 108 cm

AD = \(\sqrt{108}\)

Hence, the height of the triangle is \(6\sqrt{3}\)

Question 10:Find the length of a diagonal of a rectangle whose adjacent sides are 30 cm and 16 cm.

Solution:

Let ABCD is the given rectangle, let BD is a diagonal making a \(\Delta\)

\(\Rightarrow\)^{o}

Using Pythagoras theorem:

(DB)^{2} = AB^{2} + AD^{2}

DB^{2} = (16^{2} + 30^{2}) cm^{2}

DB = \(\sqrt{16^{2}+30^{2}}\)

Hence, length of diagonal DB is 34 cm.

Question 11:Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long.

Solution:

Let ABCD be the given rhombus whose diagonals intersect at O.

Then AC = 24 cm and BD = 10 cm

We know that the diagonals of a rhombus bisect each other at right angles.

OA = ½ AC = 12 cm

OB = ½ BD = 5 cm

And \(\angle\)^{o}

From right \(\Delta\)

AB^{2} = OA^{2} + OB^{2}

\(\Rightarrow\)^{2} = [(12)^{2} + (5)^{2}] cm^{2} = (144 + 25) cm^{2} = 169 cm^{2}

\(\Rightarrow\)

Hence, each side of a rhombus is 13 cm.

Question 12:In \(\Delta\)

AB^{2} = AD^{2} – BC.DE + \(\frac{1}{4}\)^{2}.

Solution:

Given:\(\Delta\)

Then, BD = CD and \(\angle\)^{o}

Then, \(\angle\)^{o} and \(\angle\)^{o}

In \(\Delta\)

\(\angle\)^{o}

Therefore, AD^{2} = AE^{2} + DE^{2}

\(\Rightarrow\)^{2} = (AD^{2} – DE^{2}) ——- (1)

In \(\Delta\)

\(\angle\)^{o}

Therefore, AB^{2} = AE^{2} + BE^{2} ——- (2)

Putting the value of AE^{2} from (1) in (2), we get

Therefore, AB^{2} = (AD^{2} – DE^{2}) + BE^{2}

= (AD^{2} – DE^{2}) + (BD^{2} – DE^{2}) [But BD = ½ BC]

= AD^{2} – DE^{2} + \(\left ( \frac{1}{2}BC-DE \right )^{2}\)

= \(AD^{2}-DE^{2}+\frac{1}{4}BC^{2}+DE^{2}-BC.DE\)

AB^{2} = AD^{2} – BC.DE + \(\frac{1}{4}\)^{2}

Question 13: In the given figure, D is the midpoint of side BC and AE \(\perp\)

(i) b^{2} = p^{2} + ax + \(\frac{a^{2}}{4}\)

(ii) c^{2} = p^{2} – ax + \(\frac{a^{2}}{4}\)

(iii) (b^{2} + c^{2}) = 2p^{2} + \(\frac{1}{2}\)^{2}

(iv) (b^{2} – c^{2}) = 2ax

Solution:

Given: D is the midpoint of side BC, AE \(\perp\)

In \(\Delta\)^{o}

AD^{2} = 2AE^{2} + ED^{2} (by Pythagoras theorem)

\(\Rightarrow\)^{2} = h^{2} + x^{2}

(i)In \(\Delta\)^{o}

\(b^{2}=h^{2}+\left ( x+\frac{a}{2} \right )^{2}=(h^{2}+x^{2})+ax+\frac{a^{2}}{4}\)

Therefore, b^{2} = p^{2} + ax + \(\frac{a^{2}}{4}\)

(ii)In \(\Delta\)^{o}

AB^{2} = AE^{2} + BE^{2} (by pythagoras theorem)

\(\Rightarrow\)

= \((h^{2}+x^{2})-ax+\frac{a^{2}}{4}\)

= \(p^{2}-ax+\frac{a^{2}}{4}\)

Hencxe, c^{2} = p^{2} – ax + \(\frac{a^{2}}{4}\)

(iii)Adding (1) and (2), we get:

\(b^{2}+c^{2}=p^{2}+ax+\frac{a^{2}}{4}+p^{2}-ax+\frac{a^{2}}{4}\)

\((b^{2}+c^{2})=2p^{2}+\frac{1}{2}a^{2}\)

(iv)Subtracting (2) from (1), we get:

\(b^{2}-c^{2}=p^{2}+ax+\frac{a^{2}}{4}-\left ( p^{2}-ax+\frac{a^{2}}{4} \right )\)

= \(p^{2}+ax+\frac{a^{2}}{4}-p^{2}+ax-\frac{a^{2}}{4}\)

\((b^{2}-c^{2})=2ax\)

Question 14:In \(\Delta\)

(AD^{2} – AC^{2}) = BD.CD.

Solution:

Const:Draw a perpendicular AE from A

Thus, AE \(\perp\)

Proof:

In \(\Delta\)

And AE is a bisector of BC

Then, BE = EC

In right angle triangles \(\Delta\)

AD^{2} = AE^{2} + DE^{2} —— (1)

AC^{2} = AE^{2} + CE^{2} ——- (2)

Subtracting (2) from (1),

\(\Rightarrow\)^{2} – AC^{2}) = DE^{2} – CE^{2}

\(\Rightarrow\)^{2} – AC^{2} = (DE + CE) (DE – CE) = (DE + BE) (DE – CE) [CE = BE]

\(\Rightarrow\)^{2} – AC^{2} = BD x CD

Hence proved