Question

A tangent to a circle at point A is given, and point A is an endpoint of a chord, which is the same length as radius of the circle. What is the measure of angle between the tangent and the chord?
PLS HELP SQDANCEFAN!!!!! I NEED HELP I DON'T GET IT :((((

Answer 1

Answer:

30°

Step-by-step explanation:

Call the other end of the chord point B and the center of the circle point O. Then triangle AOB is an equilateral triangle, since OA = OB = AB.

Angle OAB is the internal angle of that triangle, so is 60°. Since OA is perpendicular to the tangent line (makes an angle of 90°), The angle between the tangent line and the chord must be ...

90° - 60° = 30°

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The other way you know this is that central angle AOB is 60°, and the tangent-to-chord angle is half that, or 30°.

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One way to remember the angle relationship between a tangent line and a chord is this:

Consider a point C on long arc AB. The measure of inscribed angle ACB is half the measure of central angle AOB, no matter where C is on the circle. (If C happens to be on the short arc AB, then central angle AOB is a reflex angle, but the relationship still holds.) Consider what happens when C approaches A. The angle at vertex C is still the same: 1/2 the measure of central angle AOB. This remains true even in the limit when points A and C become coincident and line AC is a tangent at point A.