Question

In the picture, line RS is tangent to circle C at point R. The measure of ∠SRY is 40°.

Use the drop-down menus to answer the questions below.

Answer 1

m∠SRC = 90°

m∠YRC = 50°

m∠RYC = 50°

m∠YCR = 80°

m arcRZY = 80°

Explanation:

A tangent to a circle is always perpendicular to the radius at the point of tangency. Hence ∠SRC = 90°.

∠YRC is the difference between ∠SRC and ∠SRY, which is given as 40°. Hence ∠YRC is 90°-40° = 50°.

ΔYCR is isosceles, to ∠RYC = ∠YRC = 50°.

The sum of angles in a triangle is 180°, so the central angle YCR can be found from ...

∠YCR + 50° + 50° = 180°

∠YCR = 80°

The problem statement tells you the arc measure is the same as the angle you just found, so is 80°.

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Comment on the question

In the end, the arc measure (and the corresponding central angle) are double the angle made by the tangent and the chord RY. You can use a variable for the value of ∠SRY and work through this again to verify that will always be the case.