In the given figure, AB and AC are the tangents to the circle with centre O, which are drawn from an external point A. If AB = 20 cm and BC = 24 cm, then what is the circumference of the circle?

In the given figure, AB and AC are the tangents to the circle with centre O, which are drawn from an external point A. If AB = 20 cm and BC = 24 cm, then what is the circumference of the circle? 15:37 @ 0.76K/S O Jio 4G 4G VoLTE CellOne 3G (Ä74% Topic Test Use the following information to answer the next question. In the given figure, AB and AC are the tangents to the circle with centre O, which are drawn from an external point A. •o c If AB = 20 cm and BC = 24 cm, then what is the circumference of the circle? 26Tt cm Submit

Dear Student,

Please find below the solution to the asked query:

We form our diagram , As :

Here , OM is perpendicular on BC and we know perpendicular from center to a chord also bisect that chord , So

BM  =  CM  =  12 cm                       ( Given BC  =  24 cm )

Now we apply Pythagoras theorem in triangle ABM and get

AB² = BM² + AM² , Substitute values we get

20² =  12² + AM²  ,

400 = 144 + AM² ,

AM² = 256 ,

AM² =  16² ,

AM = 16 cm

Now we apply Pythagoras theeorem in triangle OBM and get

OB² = BM² + OM² , Substitute values we get

OB² =  12² + OM²  ,

OB² =  144 + OM²                        --- ( 1 )

We know " A tangent to a circle is perpendicular to the radius at the point of tangency. " So

$\angle \mathrm{OBA} = 90°$ , Then we apply Pythagoras theeorem in triangle OAB and get

OA² = OB² + AB² , Substitute values we get

( AM + OM )² =  OB² + 20²  ,

( 16 + OM )² =  OB² + 20²  ,       As we get above ( AM = 16 cm )

256 + OM² + 32 OM =  OB² + 400 ,

OB² = OM² + 32 OM - 144 , Now we substitute values from equation 1 and get

144 + OM² = OM² + 32 OM - 144

32 OM  =  288

OM = 9 , Substitute that value in equation 1 and get

OB² = 144 + 9² ,

OB² = 144 + 81 ,

OB² = 225 ,

OB² = 15² ,

OB =  15  cm , So radius of given circle is =  15  cm    ( As OB is radius of given circle )

We know circumference of circle  = 2 $\mathrm{\pi }$r  , So

Circumference of given circle  = $2 \times \frac{22}{7}\times 15 = \frac{660}{7}$ = 94.28 cm                                     ( Ans )

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