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State True or False.
In the following figure, D is a point on the side BC of a triangle ABC such that $∠ ADC = ∠ BAC$ . Then $CA ² = CB ⋅ CD$ .

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True

False

answer is A.

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Detailed Solution

Given that D is a point on the side BC of a triangle ABC such that

∠ ADC = ∠ BAC

Δ ABC and Δ DAC

∠ ADC = ∠ BAC

[given]

∠ DCA = ∠ ACB

[common angles]
By AA similarity

, Δ ABC ~ Δ DAC

.
We know that if two triangles are similar, then their corresponding sides are in proportion.
Using properties of similar triangles,

A C D C = C B C A

⇒ CA ² = CB ⋅ CD

The answer is the same as the statement.

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