Question

Given two equal chords AB and CD of a circle with centre o intersecting each other at point p. Prove that AP = CP and BP = DP

Answer 1

equation (i) ---------- given

equation (ii) ----------- if two chords intersect each other then product of their segments are equal

Answer 2

Step-by-step explanation:

First we have to prove that

∆PAD~∆PCD.

In ∆PAD and ∆PCB,

we have

$\angle PAD = \angle PCB$

/* Angle in the same segment of arc BD */

$\angle APD = \angle CPB$

/* Vertically opposite angles */

So, by AAA-criterian of similarity, we have

∆PAD ~ ∆PCB

$\implies \frac{PA}{PC}=\frac{PD}{PB}$

/* Corresponding sides of similar triangles are in the same ratio */

$\implies \frac{AP}{CP}=\frac{PD}{PB}$

$1=\frac{PD}{PB}$

$\implies PB=PD---(1)$

$Since,AP=CP$

$AP=CP--(2)\\ \:\implies \frac{AP}{CP}=1$

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