Tangent to a Circle Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Tangent to a Circle.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The defining property of a tangent line is that it is perpendicular to the radius at the point of contact. This right angle (\text{tangent} \perp \text{radius}) is the key to solving nearly every tangent-line problem.

Common stuck point: Two tangent lines from an external point to a circle are always equal in length. This is a powerful problem-solving tool.

Sense of Study hint: When you see a tangent-to-circle problem, first draw the radius to the point of tangency. Then mark the right angle between the radius and the tangent line. Finally, use the Pythagorean theorem or properties of the right triangle formed.

Worked Examples

Example 1

easy
A 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.

Solution

  1. 1
    Step 1: Recall that a tangent to a circle is perpendicular to the radius at the point of tangency. So OP \perp AP, making triangle OAP a right triangle with the right angle at P.
  2. 2
    Step 2: Identify the hypotenuse: OA = 25 cm (from center to external point), and one leg OP = 7 cm (radius).
  3. 3
    Step 3: Apply the Pythagorean theorem: AP^2 + OP^2 = OA^2, so AP^2 = 25^2 - 7^2 = 625 - 49 = 576.
  4. 4
    Step 4: Take the square root: AP = \sqrt{576} = 24 cm.

Answer

AP = 24 cm
Because a tangent is perpendicular to the radius at the point of tangency, triangle OAP is right-angled at P. This is a classic 7-24-25 Pythagorean triple, giving AP = 24 cm.

Example 2

medium
Two 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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Line \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 2

hard
A 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.
← Back to Tangent to a Circle Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

circlesperpendicularity