Practice Tangent to a Circle in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A line that touches a circle at exactly one point, called the point of tangency. At this point, the tangent line is perpendicular to the radius.
Imagine a ball sitting on a flat floor. The floor touches the ball at exactly one point—that's tangency. The floor (tangent line) is perfectly perpendicular to a line from the ball's center to the contact point (the radius). No matter how you tilt the flat surface, if it only touches at one point, it must be perpendicular to the radius there.
Example 1
easyA tangent line touches circle O at point P. The radius OP = 7 cm. A line from an external point A is tangent to the circle at P, and OA = 25 cm. Find the length of the tangent segment AP.
Example 2
mediumTwo tangent segments PA and PB are drawn from external point P to circle O. If PA = 3x - 4 and PB = x + 8, find the lengths of both tangent segments.
Example 3
easyLine \ell is tangent to circle O at point T. If the radius OT = 5 and a point A on line \ell satisfies OA = 13, find AT.
Example 4
hardA circle with center O and radius r = 6 is inscribed in angle \angle BAC (i.e., tangent to both rays AB and AC). The tangent points are D on AB and E on AC. If AD = 9, find AE, and then find the length AO.