Question

Circle R has a radius of line segment of QR and QP is a tangent to circle R at point Q.

(Picture attached)

(a) What is the measure of RQP? Explain your answer.

(b) What is the value of x? Explain your answer with work.

(c) What is the measure of QRP? Explain your answer with work.

(d) What is the measure of RPQ? Explain your answer with work.

Answer 1

(a) 90°

(b) 8.75

(c) 63.75°

(d) 26.25°

Explanation:

(a) A radius to a point of tangency is always perpendicular to the tangent line there. Q is the point of tangency of line PQ, so the segment RQ from the center of the circle, R, to that point makes a 90° angle with PQ. Angle RQP is 90°.

(b) The sum of the acute angles of a right triangle is 90°, so ...

(5x +20)° + (3x)° = 90° . . . . . the sum of the acute angles is 90°

8x + 20 = 90 . . . . . . . . . . . . simplify, divide by °

8x = 70 . . . . . . . . . . . . . . . . . subtract 20

70/8 = x = 8.75 . . . . . . . . . . . divide by the coefficient of x

(c) ∠QRP = (5x+20)° = (5·8.75 +20)° = 63.75° . . . . . use the value of x in the expression for the angle measure

(d) ∠RPQ = (3x)° = (3·8.75)° = 26.25° . . . . . use the value of x in the expression for the angle measure