Question

Two tangents make an angle of 120 degree with each other are drawn to a circle of radius 6 CM show that the length of each tangent is 2✓3.

plz answer this question....​

Answer 1

${\underline{\underline{\frak{Answer\;:}}}}$

$\setlength{\unitlength}{1.1mm}\begin{picture}(50,55)\thicklines\qbezier(25.000,10.000)(33.284,10.000)(39.142,15.858)\qbezier(39.142,15.858)(45.000,21.716)(45.000,30.000)\qbezier(45.000,30.000)(45.000,38.284)(39.142,44.142)\qbezier(39.142,44.142)(33.284,50.000)(25.000,50.000)\qbezier(25.000,50.000)(16.716,50.000)(10.858,44.142)\qbezier(10.858,44.142)( 5.000,38.284)( 5.000,30.000)\qbezier( 5.000,30.000)( 5.000,21.716)(10.858,15.858)\qbezier(10.858,15.858)(16.716,10.000)(25.000,10.000)\put(25,30){\line(1,3){6.4}}\put(25,30){\line(1, - 3){6.4}}\put(20,7){\line(3,1){68.5}}\put(20,52.8){\line(3, - 1){68.5}}\put(25,30){\circle*{1}}\put(88,30){\circle*{1.2}}\put(22,30){\sf O}\put(31,6){\sf A}\put(31,52){\sf B}\put(90,28.5){\sf P}\multiput(25,30)(3,0){21}{\line(2,0){1.8}}\qbezier(81.5,32)(78,30)(81,27.5)\qbezier(76,30)(74.5,28)(76,26)\put(33,45){\line(1,3){1.1}}\put(30,45.8){\line(3, - 1){3.2}}\put(30.5,14){\line(3,1){3.2}}\put(33.5,15){\line(1, - 3){1.1}}\put(21,18){\sf 6 cm}\put(73,30.8){\sf 120^\circ }\put(69,26){\sf 60^\circ }\end{picture}$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

To Prove:

• AP = BP = 2√3 cm

⠀

Proof:

⠀

In ∆ OAP,

⠀

$\sf\angle\;OPA = 60^\circ$

$:\implies\sf Sin\;60^\circ = \dfrac{OA}{OP}$

$:\implies\sf \dfrac{ \sqrt{3}}{2} = \dfrac{6}{OP}$

$:\implies\sf OP = \dfrac{12}{ \sqrt{3}} \times \dfrac{ \sqrt{3}}{ \sqrt{3}}$

$:\implies\sf OP = \dfrac{ 12 \sqrt{3}}{3}$

$:\implies\sf{\boxed{\frak{\pink{OP = 4 \sqrt{3}\;cm}}}}\;\bigstar$

$\underline{\sf{\bigstar\;Now,\;Using\; Pythagoras\; Theorem\;:}}$

$:\implies\sf OP^2 = OA^2 + AP^2$

$:\implies\sf AP^2 = OP^2 - OA^2$

$:\implies\sf AP^2 = ( 4 \sqrt{3})^2 - (6)^2$

$:\implies\sf AP^2 = 48 - 36$

$:\implies\sf AP^2 = 12$

$:\implies\sf \sqrt{AP^2} = \sqrt{12}$

$:\implies\sf{\boxed{\frak{\purple{AP = 2 \sqrt{3}\;cm}}}}\;\bigstar$

$\qquad\qquad\quad\dag\;{\underline{\underline{\bf{\pink{Hence,\;Proved!!}}}}}$