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Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If AC and BD intersect at P, then which of the following is true?
Option: 1
$A P \times P C=B P \times P D$

Option: 2
$A P \times P D=B P \times P C$

Option: 3
$A P \times P C\neq B P \times P D$

Option: 4
None of the above

Answers (1)

$\\\text{In } \triangle B A P\text{ and }\triangle C D P,\text{ we have} \\ \\\angle B A P=\angle C D P=90^{\circ} \\ \\\angle B P A=\angle C P D \quad \text { (ver. opp. } \angle_S) \\\\ \therefore \quad \triangle B A P \sim \triangle C D P \quad[\text { by AA-similarity }]\\\\\therefore \quad \frac{A P}{D P}=\frac{B P}{C P}$

$\Rightarrow \quad A P \times C P=B P \times D P \Rightarrow A P \times P C=B P \times P D$

Posted by

Sumit Saini

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