Tangent to a Circle Examples: 24 Problems with Answers

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: A tangent line touches a circle at exactly one point and is perpendicular to the radius drawn to that point.

Common stuck point: The procedure for tangent to a circle is the easy part; the trap is forgetting the right angle. Asking "Does the line meet the circle at exactly one point, making it perpendicular to the radius there?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the line meet the circle at exactly one point, making it perpendicular to the radius there?

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.

Answer

AP=24 cm

First step

1
Step 1: Recall that a tangent to a circle is perpendicular to the radius at the point of tangency. So OP⊥AP, making triangle OAP a right triangle with the right angle at P.

Full solution

  1. 2
    Step 2: Identify the hypotenuse: OA=25 cm (from center to external point), and one leg OP=7 cm (radius).
  2. 3
    Step 3: Apply the Pythagorean theorem: AP2+OP2=OA2, so AP2=252−72=625−49=576.
  3. 4
    Step 4: Take the square root: AP=576=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.

Example 3

medium
A circle has center O(0,0) and radius 5. Verify that the line x=5 is tangent to the circle and identify the point of tangency.

Example 4

medium
A circle is inscribed in a triangle with sides 5, 12, 13. Find the radius of the inscribed circle (incircle).

Example 5

hard
External point P has tangent length 8 to circle O. A secant from P enters the circle and the near intersection has PB=4. Find the length of the secant chord inside the circle (BC).

Example 6

hard
A circle has center (2,3) and radius 5. Determine whether the line 3x+4y=43 is tangent to the circle.

Example 7

challenge
A tangent line at T to a circle of radius r centered at O makes an acute angle θ with chord TA. If arc TA (the arc on the same side as the angle) measures 80°, find θ.

Practice Problems

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

Example 1

easy
Line ℓ is tangent to circle O at point T. If the radius OT=5 and a point A on line ℓ satisfies OA=13, find AT.

Example 2

hard
A circle with center O and radius r=6 is inscribed in 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.

Example 3

easy
A tangent line touches circle O at T. If OT=6 and OP=10 for external point P on the tangent, find PT.

Example 4

easy
A radius drawn to the point of tangency has length 7. From an external point P, OP=25. Find the tangent length PT.

Example 5

easy
A circle has radius 9. From external point P, the tangent length to the circle is 12. Find OP.

Example 6

medium
Tangent segments PA and PB are drawn from external point P. If PA=2x+3 and PB=5x−12, find PA.

Example 7

medium
From external point P, tangents PA and PB are drawn to circle O with ∠APB=40°. Find ∠AOB.

Example 8

medium
A common external tangent touches two circles of radii 4 and 4 whose centers are 12 apart. Find the length of the tangent segment between the tangent points.

Example 9

medium
Tangent PT to circle O has length 15 and external point P satisfies OP=17. Find the radius.

Example 10

medium
Tangents from external point P touch a circle at A and B, with ∠APB=60°. Find ∠OAP.

Example 11

medium
A tangent-chord angle measures 35°. Find the measure of the intercepted arc.

Example 12

hard
From external point P, PA is tangent to circle O at A, and PBC is a secant where PB=4 and PC=9. Find PA.

Example 13

hard
Two circles with centers 4 apart have radii 1 and 2. Find the length of a common external tangent segment between the tangent points.

Example 14

hard
Two circles with centers 10 apart and radii 3 and 5. Find the length of a common internal tangent.

Example 15

hard
From external point P, two tangents to circle O make an angle of ∠APB=90° at P. If the radius is r=6, find OP.

Example 16

hard
A circle of radius r is inscribed in a square of side 10. The circle is tangent to all four sides. Find r.

Example 17

challenge
Triangle ABC has sides a=13, b=14, c=15, all tangent to its incircle. The tangent lengths from A (to its two touching points) are equal to s−a. Find this tangent length.
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Background Knowledge

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

circlesperpendicularity