Practicing Success
If two tangents to a circle of radius 3 cm are inclined to each other at angle of 60°, then the length of each tangent is: |
$\frac{3\sqrt{3}}{4}$cm $3\sqrt{3}$cm 3 cm 6 cm |
$3\sqrt{3}$cm |
Let AP and CP be the tangents drawn to a circle According to the question, \(\angle\)CPA = \({60}^\circ\) So, AP = CP In triangle PAO and PCO, = AP = CP = AP = CO = radius of circle = PO = PO = common side Triangle PAO and triangle PCO are similar, by SSS criterion, At the point of contact the radius of a circle is perpendicular to the tangent. So, \(\angle\)PAO = \(\angle\)PCO = \({90}^\circ\) Since AO = CO = radius = \(\angle\)PAO = \(\angle\)PCO = \(\angle\)CPA = \(\angle\)PAO + \(\angle\)PCO = 2\(\angle\)PAO = \({60}^\circ\) = \(\angle\)PAO = \({60}^\circ\)/2 = \(\angle\)PAO = \({30}^\circ\) In triangle PAO PAO is aright angle triangle with A at right triangle. = tan \({30}^\circ\) = \(\frac{AO}{PA}\) = \(\frac{1}{√3}\) = \(\frac{3}{PA}\) = AP = 3√3 cm. Therefore, length of each tangent is 3√3 cm. |