Question

Ey (2) Point O is the centre of a circle. Line a and line b are parallel tangents to the
circle at P and Q. Prove that segment PQ is a diameter of the circle.
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Answer 1

Answer:

Given: AB⊥PQ

AX is a tangent at A,AX⊥PX

PQ is a diameter

In △PAQ

∠PAQ=90∘

Since AX is a tangent

According to the alternate segment theorem, angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

∠XAP=∠AQP–(1)

∠APQ+∠PQA+∠QAP=180

∠APQ=90−∠PQA

∠APQ=90−∠XAP–(2)

In △AXP

∠PXA=180–90−∠XAP

∠PXA=90−∠XAP

From (2) and (3)

∠APQ=∠PXA

In triangles AXPandPXA

∠APQ=∠PXA

∠AXP=∠ANP=90

AP=AP (Common)

By RHS congruency △AXP≅△ANP

By CPCT,

PN=PX

Answer 2

Answer:

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