Question

Assertion: In the given figure AB = 12 cm, CD = 13 cm and AD = 11 cm. ∴ AC = 20 cm.

Reason: In ΔABC, if ∠B = 90°, then AB² + BC² = AC².

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Step 1: Check the Assertion:

From the figure:

AB = 12 cm   ...(Vertical)

AD = 11 cm and AD || BC  ...(both horizontal)

DC = 13 cm

∠B = 90°

Take coordinates:

B(0, 0), so A(0, 12)

Let BC = x, so C(x, 0)

Then D(11, 12)

Use CD = 13 in △CDA:

CD² = (x − 11)² + (0 − 12)² = 13²

(x − 11)² + 144 = 169

(x − 11)² = 25

x − 11 = 5

x = 16

So BC = 16 cm

AC = 20 cm

Now in △ABC:

AC² = AB² + BC²

= 12² + 16²

= 144 + 256

= 400

AC = `sqrt(400)`

AC = 20 cm

So Assertion is true.

Step 2: Check the Reason:

In ΔABC, if ∠B = 90°, then AB² + BC² = AC².

This is exactly the Pythagoras theorem, and it is correct and is the method used to get AC.

So Reason is true and is the correct reason for Assertion.