Question

XY and XZ are two tangents from an external point X to a circle with centre C.
Prove that XY = XZ.

Answer 1

Step-by-step explanation:

In △AOP & △AOC

OP=OC                          [Both radius]

AP=AC                          [Length of tangents drawn from external point to a circle are equal]

OA=OA                          [common]

∴△AOC≅△AOP       [SSS Congruence rule]

So, ∠AOP=∠AOC...............(1)                          [CPCT]

Now,

In △BOC & △BOQ

OC=OQ                         [Both radius]   

BC=BQ                          [Length of tangents drawn from external point to a circle are equal]

OB=OB                          [common]

∴△BOC≅△BOQ       [SSS Congruence rule]

So, ∠BOC=∠BOQ...............(2)                          [CPCT]

For line PQ

∠AOP+∠AOC+∠BOC+∠BOQ=180

∠AOC+∠AOC+∠BOC+∠BOC=180

2∠AOC+2∠BOC=180

2(∠AOC+∠BOC)=180

∠AOC+∠BOC=2180

∠AOC+∠BOC=90

∠AOB=90

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