Question

In the given figure, ∠BAC = 90° and AD ⊥ BC. Then,

Options

• BC · CD = BC²

• AB · AC = BC²

• BD · CD = AD²

• AB · AC = AD²

Answer 1

BD · CD = AD²

Explanation:

1. Identifying similar triangles

In ΔABC, we are given that ∠BAC = 90° and AD ⊥ BC.

Let ∠BCA = θ.

In the right-angled ΔABC:

∠ABC = 90° – θ

In the right-angled ΔADC (where ∠ADC = 90°):

∠DAC = 90° – θ

In the right-angled ΔADB (where ∠ADB = 90°):

∠BAD = 90° – (90° – θ) = θ 

By comparing the angles of ΔDBA and ΔDAC, we have:

1. ∠ADB = ∠ADC = 90°

2. ∠ABD = ∠DAC = 90° – θ

3. ∠BAD = ∠ACD = θ

By AA (Angle-Angle) similarity criterion, the two smaller triangles are similar to each other:

ΔDBA ∼ ΔDAC

2. Setting up the ratio

Since corresponding sides of similar triangles are in the same proportion, we can relate their sides:

`(BD)/(AD) = (AD)/(CD)`

3. Deriving the final relationship

By cross-multiplying the proportional side lengths from the previous step:

BD · CD = AD · AD

BD · CD = AD²