Question

Assertion: In the given figure in ΔABC, ∠B = 90° and DE ⊥ BC. AB = 20 cm, AC = 25 cm = CD. BE = 8 cm, then y = 24 cm.

Reason: BC = 15 cm ∴ x = 7 cm.

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 but R is the incorrect reason for A.

Explanation:

Step 1: Verification of Assertion:

Calculation of BC:

In ΔABC, ∠B = 90°.

Given AB = 20 cm and AC = 25 cm

Therefore, 20² + BC² = 25²

400 + BC² = 625

BC² = 625 – 400

BC² = 225

BC = `sqrt(225)`

BC = 15 cm

Calculation of CE:

It is given that CD = 25 cm.

CE = CD – DE. However, DE is not directly given.

Alternatively, CE = BC – DE

Given BE = 8 cm and BC = 15 cm

Therefore, CE = 15 – 8 = 7 cm

Calculation of y (DE): 

Given CD = 25 cm and CE = 7 cm

In ΔCDE, ∠E = 90°  .....(since DE ⊥ BC)

By the Pythagorean theorem, 

DE² + CE² = CD²

DE² + 7² = 25²

DE² + 49 = 625

DE² = 625 – 49

DE² = 576

DE = `sqrt(576)`

DE = 24 cm

Since y = DE, it is found that y = 24 cm

Thus, the assertion is true.

Step 2: Verification of Reason:

Evaluation of BC: As calculated in Step 1, BC = 15 cm.

The reason states BC = 15 cm. 

This part of the reason is true.

Step 3: Conclusion on the relationship between Assertion and Reason:

Both the assertion and the reason are found to be true statements.

However, the reason does not provide a correct or complete explanation for the assertion. The calculation of BC is a step in verifying the assertion, but the reason’s conclusion, ∴ 7 cm is vague and does not explain why y = 24 cm.