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D is a point on the side BC of ∆ABC such that ∠ADC = ∠BAC. Prove that CA/CD=CB/CAor, CA^2 = CB × CD. - Mathematics

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D is a point on the side BC of ∆ABC such that ∠ADC = ∠BAC. Prove that ` \frac{CA}{CD}=\frac{CB}{CA} or, CA^2 = CB × CD.`

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Solution 1

In ∆ABC and ∆DAC, we have

∠ADC = ∠BAC and ∠C = ∠C

Therefore, by AA-criterion of similarity, we have

∆ABC ~ ∆DAC

`\Rightarrow \frac{AB}{DA}=\frac{BC}{AC}=\frac{AC}{DC}`

`\Rightarrow \frac{CB}{CA}=\frac{CA}{CD}`

Solution 2

In ΔADC and ΔBAC,

∠ADC = ∠BAC (Given)

∠ACD = ∠BCA (Common angle)

∴ ΔADC ∼ ΔBAC (By AA similarity criterion)

We know that corresponding sides of similar triangles are in proportion.

`:. (CA)/(CB) = (CD)/(CA)`

=> CA2 = CB x CD

Concept: Criteria for Similarity of Triangles
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APPEARS IN

NCERT Class 10 Maths
Chapter 6 Triangles
Exercise 6.3 | Q 13 | Page 141
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