Assessment: Circles

What angle does a tangent line always make with the radius at the point of tangency?

Two tangents are drawn to a circle from the same external point. How do their lengths compare?

A circle has radius 9. A tangent from external point P touches at T, with OP = 15. Find PT.

A tangent-chord angle measures 73°. Find the inscribed angle in the alternate segment.

Tangents from external point R touch a circle at S and T. RS = 4y − 3 and RT = 2y + 7. Find y.

In quadrilateral ABCD, ∠A = 88° and ∠C = 92°. Are A, B, C, D concyclic?

A cyclist rides in a straight line that just touches a circular roundabout at one point. What angle does this path make with the roundabout's radius at that point?

A surveyor measures the angle two different observation points make to a fixed baseline and finds both angles are exactly 40°, with both observers on the same side. What can be concluded?

Why does the tangent-chord angle equal the inscribed angle in the alternate segment (not the same segment)?

A circle has an unknown radius r. A tangent from external point P (OP=17) touches the circle at T, with PT=15. Find r.