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.

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

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.

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

$\Rightarrow 90 ° + ∠ A O B = 180 °$

$\Rightarrow ∠ A O B = 180 ° - 90 °$

$\Rightarrow ∠ A O B = 90 °$

Want Us to Email you About Special Offers & Updates?