Mensuration — Assertion-Reason Type Questions

Assertion (A): Sides of a triangle are 9 cm, 12 cm and 15 cm. This triangle is both scalene triangle and right angle triangle.

Reason (R): Area of a right triangle = $\dfrac{1}{2}$ x base x height.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Given, sides of a triangle are 9 cm, 12 cm and 15 cm.

A scalene triangle is a triangle in which all three sides have different lengths.

The given side lengths are 9 cm, 12 cm, and 15 cm. All these lengths are distinct.

Therefore, it is a scalene triangle.

If the square of the longest side equals the sum of the squares of the other two sides, then it's a right-angled triangle.

The longest side is 15 cm.

Let's check: 15² = 9² + 12²

⇒ 225 = 81 + 144

⇒ 225 = 225

Since the equality holds, the triangle is a right-angled triangle.

∴ Assertion (A) is true.

By formula,

Area of a right triangle = $\dfrac{1}{2}$ x base x height.

∴ Reason (R) is true.

∴ Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Hence, option 4 is the correct option.

Assertion (A): Heron's formula can be used to find the area of a scalene triangle only.

Reason (R): If ABC is a triangle with side a, b and c respectively, then its area = $\sqrt{s(s - a)(s - b)(s - c)}$, where s = $\dfrac{a + b + c}{2}$.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

With the help of heron's formula, we can find the area of any triangle and not particularly scalene triangle.

∴ Assertion (A) is false.

If ABC is a triangle with side a, b and c respectively, then its area = $\sqrt{s(s - a)(s - b)(s - c)}$, where s = $\dfrac{a + b + c}{2}$.

This is the correct definition and formula for Heron's formula. The variable 's' represents the semi-perimeter of the triangle.

∴ Reason (R) is true.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): The volume of a cuboid having length, breadth and diagonal as 4 m, 3 m and 13 m is 144 m³.

Reason (R): Length of diagonal of a cuboid is $\sqrt{l^2 + b^2 + h^2}$.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Given: length, breadth and diagonal of cuboid = 4 m, 3 m and 13 m.

By formula,

Length of diagonal of a cuboid = $\sqrt{l^2 + b^2 + h^2}$

∴ Reason (R) is true.

Substituting the values, we get

$\Rightarrow 13 = \sqrt{3^2 + 4^2 + h^2}\\[1em] \Rightarrow 13^2 = 3^2 + 4^2 + h^2\\[1em] \Rightarrow 169 = 9 + 16 + h^2\\[1em] \Rightarrow 169 = 25 + h^2\\[1em] \Rightarrow h^2 = 169 - 25\\[1em] \Rightarrow h^2 = 144\\[1em] \Rightarrow h = \sqrt{144}\\[1em] \Rightarrow h = 12$

Volume of cuboid = l x b x h

= 3 x 4 x 12

= 144 m³.

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.