High School

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In a circle with center O, if tangent QP is drawn at point P on the circle and triangle TOP is isosceles, what is the measure of angle OQP?

Answer :

Final answer

In a circle with center O, if tangent QP is drawn at point P on the circle and triangle TOP is isosceles, the measure of angle OQR is equal to the measure of angle TOP.

Explanation:

In a circle with center O, if tangent QP is drawn at point P on the circle and triangle TOP is isosceles, the measure of angle OQR can be determined using the properties of tangents and isosceles triangles. Since triangle TOP is isosceles, we know that angles TPO and TOP are congruent. Therefore, angle TPO = angle TOP.

Since tangent QP is drawn at point P, angle QPO is a right angle. Therefore, angle QPO = 90 degrees. From the properties of a circle, we know that a tangent to a circle is perpendicular to the radius drawn to the point of contact. Therefore, angle QPO is also congruent to angle OPQ. Therefore, angle QPO = angle OPQ = angle TPO = angle TOP.

Since angles QPO and OPQ are congruent, we can conclude that angle OQR is also congruent to angle OPQ = angle QPO = angle TPO = angle TOP. So, the measure of angle OQR is equal to the measure of angle TOP.

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