D is a point on the side B C of - A B C- such that - A D C and - B A C are equal- Prove that- C A2-D C - C B

Given: In ΔABC, D is a point on BC, such that ∠ADC = ∠BAC. To prove: CA2 = DC × CB. Proof: In ΔABC and ΔDAC, we have: ∠BAC = ∠ADC nbsp; (given) ∠ACB = ∠DCA ...

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Standard X

Mathematics

Pythagoras Theorem

D is a point ...

Question

D is a point on the side BC of ΔABC, such that ∠ADC and ∠BAC are equal.
Prove that: CA² = DC × CB.

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Solution

Given: In ΔABC, D is a point on BC, such that $∠$ ADC = $∠$ BAC.
To prove: CA² = DC × CB.
Proof: In ΔABC and ΔDAC, we have:
$∠$ BAC = $∠$ ADC (given)
$∠$ ACB = $∠$ DCA (common)
∴ Δ ABC ∼ Δ DAC [ By AA similarity]
So, the sides of ΔABC and ΔDAC are proportional.
∴ $\frac{C A}{C B} = \frac{C D}{C A}$
Hence, CA² = DC × CB

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D is a point on the side BC of ΔABC, such that ∠ADC and ∠BAC are equal. Prove that: CA2 = DC × CB.

D is a point on the side BC of ΔABC, such that ∠ADC and ∠BAC are equal.
Prove that: CA² = DC × CB.