Question

Assertion (A): In $$\triangle ABC$$, $$AB=5\ \text{cm}$$, $$BC=12\ \text{cm}$$ and $$AC=13\ \text{cm}$$. Then, $$\angle B=90^{\circ}$$.

Reason (R): $$AB^{2}+AC^{2}=BC^{2}$$.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Assertion (A) is true and Reason (R) is false.

Explanation:

Step 1 – Assertion: A is true. By the converse of Pythagoras’ theorem, the angle opposite the longest side is a right angle when the square of that side equals the sum of the squares of the other two sides.

$$AB^{2} + BC^{2} = 5^{2} + 12^{2}$$

$$= 25 + 144$$

$$= 169$$

$$AC^{2} = 13^{2}$$

$$= 169$$

Therefore, $$\angle B = 90^{\circ}$$.

Step 2 – Reason: R is false. The stated equation does not hold for the given side lengths.

$$AB^{2} + AC^{2} = 5^{2} + 13^{2}$$

$$= 25 + 169$$

$$= 194\neq 144$$

$$= 12^{2}$$

$$= BC^{2}$$