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Criteria for Similarity of Triangles

In the figure...

Question

In the figure AB = BC and AD is perpendicular to CD. Prove that : AC² = 2BC. DC.

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Solution

In △ADC, AC²= AD²+ DC² ....(1)
In △ADB, AB²= AD²+ DB² ∴AD²= AB²− DB² ...(2)
In △ADC, from (1) and (2) = (AB²− DB²) + (DB + BC)² = BC²− DB²+ DB²+ BC²+ 2DB.BC (As, AB = BC)
= 2BC²+ 2DB.BC
= 2BC (BC + DB) = 2BC.DC Hence proved

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