Find the length of AP

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Solution

Q.
The discus throw is an event in which an athlete aims to throw a discus as far as possible. The athlete spins counterclockwise one and a half times within a circle before releasing the discus. Upon release, the discus travels along a tangent to the circular path of the spin.

In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and $angle A B O space equals space 30 degree. space$ PQ is parallel to OA

Based on the above information answer the following questions:

(iii) (a) Find the length of AP

1 $\frac{75 \sqrt{3}}{2} cm$

2 $\frac{75 \sqrt{4}}{2} cm$

3 $- \frac{75 \sqrt{3}}{4} cm$

4 $\frac{76 \sqrt{3}}{2} cm$

Ans.
(1)

Since the radius of circle is perpendicular to the tangent at the point of contact $∴ ∠ O A P = 90 °$

$S i n c e P Q ∥ O A \Rightarrow ∠ O A P = ∠ Q P B = 90 ° (C o r r e s p o n d i n g a n g l e s) N o w, O B = 150 c m O Q + Q B = 150 \Rightarrow 75 + Q B = 150 \Rightarrow Q B = 150 - 75 = 75 cm I n r i g h t Δ P B Q, \cos 30 ° = \frac{P B}{Q B} \Rightarrow \frac{\sqrt{3}}{2} = \frac{P B}{75} \Rightarrow P B = \frac{75 \sqrt{3}}{2} cm A P = A B - P B = 75 \sqrt{3} - \frac{75 \sqrt{3}}{2} \Rightarrow A P = \frac{75 \sqrt{3}}{2} cm$

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