D is a point on the side BC of $△$ ABC such that $∠ A D C = ∠ B A C$ . Prove that : $C A^{2} = C B \times C D$ .

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Solution

Given in ΔABC, ∠ADC = ∠BAC
Consider ΔBAC and ΔADC
∠ADC = ∠BAC (Given)
∠C = ∠C (Common angle)
∴ ΔBAC ~ ΔADC (AA similarity criterion)
$fraction numerator A B over denominator A D end fraction equals fraction numerator C B over denominator C A end fraction equals fraction numerator C A over denominator C D end fraction c o n s i d e r comma fraction numerator C B over denominator C A end fraction equals fraction numerator C A over denominator C D end fraction C B cross times C D equals C A cross times C A$

$C A^{2} = C B \times C D$ .

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Q. In the given figure, in ∆ABC, point D on side BC is such that, ∠BAC = ∠ADC. Prove that, CA² = CB × CD

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D is a point on the side BC of △ ABC such that ∠ADC=∠BAC. Prove that : CA2=CB×CD.

Given in ΔABC, ∠ADC = ∠BAC Consider ΔBAC and ΔADC ∠ADC = ∠BAC (Given) ∠C = ∠C (Common angle) ∴ ΔBAC ~ ΔADC (AA similarity criterion) CA2=CB×CD.

D is a point on the side BC of $△$ ABC such that $∠ A D C = ∠ B A C$ . Prove that : $C A^{2} = C B \times C D$ .