Question

If XY and X’Y’ are two parallel tangents of a circle with centre O and another tangent AB
with point of contact C intersecting XY at A and X’Y’ at B. Then,
(a) Prove that AOB = 90º. (3)
(b) If AC = 4 cm and radius of the circle is 3 cm, then find the length of AO

Answer 1

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In the figure XY and X'Y' are two parallel tangents to a circle with centre O and and another tangent AB with point of contact C interesting XY at A and X'Y' at B prove that ∠AOB=90

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Solution

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Consider the problem

Let us join point O to C

In ΔOPAandΔOCA

OP=OC (Radii of the same circle)

AP=AC (Tangent from point A)

AO=AO (Common side)

ΔOPA≅ΔOCA (SSS congruence criterion)

Therefore, P↔C,A↔A,O↔O

∠POA=∠COA...(1)

Similarly,

∠QOB≅∠OCB

∠QOB=∠COB..(2)

Since,POQ is a diameter of the circle, it is a straight line.

Therefore, ∠POA+∠COA+∠COB+∠QOB=180

∘

So, from equation (1) and equation (2)

2∠COA+2∠COB=180

∘

∠COA+∠COB=90

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∠AOB=90

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Answer 2

plz refer the pic for the explanation