In the figure XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C interesting XY at A and X'Y' at B, what is the measure of $∠$ AOB.

Question

Smart Achievers · Accepted Answer

(90°)

Join OC. Since, the tangents drawn to a circle from an external point are equal.

$∴$ AP = AC

In $∆$ PAO and $∆$ AOC, we have:

AO = AO [Common]

OP = OC [Radii of the same circle]

AP = AC

$\Rightarrow ∆ P A O ≅ ∆ A O C$ $[SSS Congruency]$

$∴ ∠ P A O = ∠ C A O = ∠ 1$

$∠ P A C = 2 ∠ 1$ ...(i)

Similarly $∠ C B Q = 2 ∠ 2$ ...(2)

Again, we know that sum of internal angles on the same side of a transversal is 180°.

$∴$ $∠ P A C + ∠ C B Q = 180 °$

$\Rightarrow 2 ∠ 1 + 2 ∠ 2 = 180 ° [From (1) and (2)]$

$\Rightarrow ∠ 1 + ∠ 2 = \frac{180 °}{2} = 90 °$ ....(3)

Also $∠ 1 + ∠ 2 + ∠ A O B = 180 ° [Sum of angles of a triangle]$

$\Rightarrow 90 ° + ∠ A O B = 180 °$

$\Rightarrow ∠ A O B = 180 ° - 90 °$

$\Rightarrow ∠ A O B = 90 °$

Solution

Q.
In the figure XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C interesting XY at A and X'Y' at B, what is the measure of $∠$ AOB.

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Solution

Q. In the figure XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C interesting XY at A and X'Y' at B, what is the measure of ∠AOB.

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Q.
In the figure XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C interesting XY at A and X'Y' at B, what is the measure of $∠$ AOB.