Question

Some questions and their alternative answers are given. Select the correct alternative.
(1) Out of the following which is the Pythagorean triplet?

(A) (1, 5, 10) (B) (3, 4, 5) (C) (2, 2, 2) (D) (5, 5, 2)

(2) In a right angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse?

(A) 15 (B) 13 (C) 5 (D) 12

(3) Out of the dates given below which date constitutes a Pythagorean triplet ?

(A) 15/08/17 (B) 16/08/16 (C) 3/5/17 (D) 4/9/15

(4) If a, b, c are sides of a triangle and a² + b² = c², name the type of triangle.

(A) Obtuse angled triangle (B) Acute angled triangle (C) Right angled triangle (D) Equilateral triangle

(5) Find perimeter of a square if its diagonal is $10\sqrt{2}$ cm.

(A)10 cm (B) $40\sqrt{2}$ cm (C) 20 cm (D) 40 cm

(6) Altitude on the hypotenuse of a right angled triangle divides it in two parts of lengths 4 cm and 9 cm. Find the length of the altitude.

(A) 9 cm (B) 4 cm (C) 6 cm (D) $2\sqrt{6}$ cm

(7) Height and base of a right angled triangle are 24 cm and 18 cm find the length of its hypotenuse

(A) 24 cm (B) 30 cm (C) 15 cm (D) 18 cm

(8) In ∆ABC, AB = $6\sqrt{3}$ cm, AC = 12 cm, BC = 6 cm. Find measure of ∠A.

(A) 30° (B) 60° (C) 90° (D) 45°

Answer 1

(1) (A) In the triplet (1, 5, 10),
1² = 1, 5² = 25, 10² = 100 and 1 + 25 = 26 ≠ 100

The square of the largest number is not equal to the sum of the squares of the other two numbers.

∴ (1, 5, 10) is not a pythagorean triplet.

(B) In the triplet (3, 4, 5),
3² = 9, 4² = 16, 5² = 25 and 9 + 16 = 25

The square of the largest number is equal to the sum of the squares of the other two numbers.

∴ (3, 4, 5) is a pythagorean triplet.

(C) In the triplet (2, 2, 2),
2² = 4, 2² = 4, 2² = 4 and 4 + 4 = 8 ≠ 4

The square of the largest number is not equal to the sum of the squares of the other two numbers.

∴ (2, 2, 2) is not a pythagorean triplet.

(D) In the triplet (5, 5, 2),
2² = 4, 5² = 25, 5² = 25 and 4 + 25 = 29 ≠ 25

The square of the largest number is not equal to the sum of the squares of the other two numbers.

∴ (5, 5, 2) is not a pythagorean triplet.

Hence, the correct option is (B).


(2) According to the pythagoras theorem,
Sum of the squares of the sides making right angle is equal to the square of the third side.

∴ 169 = square of the hypotenuse
⇒ Length of the hypotenuse = $\sqrt{169}$
= 13

Hence, the correct option is (B).


(3) (A) In the triplet 15/08/17,
15² = 225, 8² = 64, 17² = 289 and 225 + 64 = 289

The square of the largest number is equal to the sum of the squares of the other two numbers.

∴ 15/08/17 is a pythagorean triplet.

(B) In the triplet 16/08/16,
16² = 256, 8² = 64, 16² = 256 and 256 + 64 = 320 ≠ 256

The square of the largest number is not equal to the sum of the squares of the other two numbers.

∴ 16/08/16 is not a pythagorean triplet.

(C) In the triplet 3/5/17,
3² = 9, 5² = 25, 17² = 289 and 9 + 25 = 34 ≠ 289

The square of the largest number is not equal to the sum of the squares of the other two numbers.

∴ 3/5/17 is not a pythagorean triplet.

(D) In the triplet 4/9/15,
4² = 16, 9² = 81, 15² = 225 and 16 + 81 = 97 ≠ 225

The square of the largest number is not equal to the sum of the squares of the other two numbers.

∴ 4/9/15 is not a pythagorean triplet.

Hence, the correct option is (A).


(4) In a triangle, if the square of one side is equal to the sum of the squares of the remaining two sides, then the triangle is a right angled triangle.

Hence, the correct option is (C).


(5)



It is given that ABCD is a square.

∴ AB = BC = CD = DA = x (say)

According to Pythagoras theorem, in ∆ABD

$$\begin{gathered} {\mathrm{AB}}^2+{\mathrm{AD}}^2={\mathrm{BD}}^2 \\ \Rightarrow x^2+x^2={\left(10\sqrt{2}\right)}^2 \\ \Rightarrow 2x^2=200 \\ \Rightarrow x^2=100 \\ \Rightarrow x=\sqrt{100} \\ \Rightarrow x=10\mathrm{cm} \end{gathered}$$

Hence, the side of the square is 10 cm.

Now,
Perimeter of a square = $4\times \left(\mathrm{side}\right)$
= $4\times x$
= $4\times 10$
= $40$ cm

Hence, the correct option is (D).


(6)


We know that,
In a right angled triangle, the perpendicular segment to the hypotenuse from the opposite vertex, is the geometric mean of the segments into which the hypotenuse is divided.

$$\begin{gathered} \therefore {\mathrm{AD}}^2=\mathrm{CD}\times \mathrm{DB} \\ =4\times 9 \\ =36 \\ \Rightarrow \mathrm{AD}=6\mathrm{cm} \end{gathered}$$

Hence, the correct option is (C).


(7) According to Pythagoras theorem,

$$\begin{gathered} {\left(\mathrm{Hypotenuse}\right)}^2={\left(\mathrm{Base}\right)}^2+{\left(\mathrm{Height}\right)}^2 \\ ={\left(18\right)}^2+{\left(24\right)}^2 \\ =324+576 \\ =900 \\ \therefore \mathrm{Hypotenuse}=30 \end{gathered}$$

Hence, the correct option is (B).


(8) In ∆ABC, AB = $6\sqrt{3}$ cm, AC = 12 cm, BC = 6 cm.

AB² = ($6\sqrt{3}$)² = 108
AC² = (12)² = 144
BC² = (6)² = 36

108 + 36 = 144

In a triangle, if the square of one side is equal to the sum of the squares of the remaining two sides, then the triangle is a right angled triangle.

In a right angled triangle, if one side is half of the hypotenuse then the angle opposite to that side is 30°.
Here, BC is half of AC.

Thus, measure of ∠A is 30°

Hence, the correct option is (A).