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: 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 OO at point PP. The radius OP=7OP = 7 cm. A line from an external point AA is tangent to the circle at PP, and OA=25OA = 25 cm. Find the length of the tangent segment APAP.

Answer

AP=24AP = 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⊥APOP \perp AP, making triangle OAPOAP a right triangle with the right angle at PP.

Full solution

  1. 2
    Step 2: Identify the hypotenuse: OA=25OA = 25 cm (from center to external point), and one leg OP=7OP = 7 cm (radius).
  2. 3
    Step 3: Apply the Pythagorean theorem: AP2+OP2=OA2AP^2 + OP^2 = OA^2, so AP2=252−72=625−49=576AP^2 = 25^2 - 7^2 = 625 - 49 = 576.
  3. 4
    Step 4: Take the square root: AP=576=24AP = \sqrt{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 PAPA and PBPB are drawn from external point PP to circle OO. If PA=3x−4PA = 3x - 4 and PB=x+8PB = x + 8, find the lengths of both tangent segments.

Example 3

medium
A circle has center O(0,0)O(0,0) and radius 55. Verify that the line x=5x = 5 is tangent to the circle and identify the point of tangency.

Example 4

medium
A circle is inscribed in a triangle with sides 55, 1212, 1313. Find the radius of the inscribed circle (incircle).

Example 5

hard
External point PP has tangent length 88 to circle OO. A secant from PP enters the circle and the near intersection has PB=4PB = 4. Find the length of the secant chord inside the circle (BCBC).

Example 6

hard
A circle has center (2,3)(2, 3) and radius 55. Determine whether the line 3x+4y=433x + 4y = 43 is tangent to the circle.

Example 7

challenge
A tangent line at TT to a circle of radius rr centered at OO makes an acute angle θ\theta with chord TATA. If arc TATA (the arc on the same side as the angle) measures 80°80°, find θ\theta.

Practice Problems

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

Example 1

easy
Line ℓ\ell is tangent to circle OO at point TT. If the radius OT=5OT = 5 and a point AA on line ℓ\ell satisfies OA=13OA = 13, find ATAT.

Example 2

hard
A circle with center OO and radius r=6r = 6 is inscribed in angle ∠BAC\angle BAC (i.e., tangent to both rays ABAB and ACAC). The tangent points are DD on ABAB and EE on ACAC. If AD=9AD = 9, find AEAE, and then find the length AOAO.

Example 3

easy
A tangent line touches circle OO at TT. If OT=6OT = 6 and OP=10OP = 10 for external point PP on the tangent, find PTPT.

Example 4

easy
A radius drawn to the point of tangency has length 77. From an external point PP, OP=25OP = 25. Find the tangent length PTPT.

Example 5

easy
A circle has radius 99. From external point PP, the tangent length to the circle is 1212. Find OPOP.

Example 6

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

Example 7

medium
From external point PP, tangents PAPA and PBPB are drawn to circle OO with ∠APB=40°\angle APB = 40°. Find ∠AOB\angle AOB.

Example 8

medium
A common external tangent touches two circles of radii 44 and 44 whose centers are 1212 apart. Find the length of the tangent segment between the tangent points.

Example 9

medium
Tangent PTPT to circle OO has length 1515 and external point PP satisfies OP=17OP = 17. Find the radius.

Example 10

medium
Tangents from external point PP touch a circle at AA and BB, with ∠APB=60°\angle APB = 60°. Find ∠OAP\angle OAP.

Example 11

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

Example 12

hard
From external point PP, PAPA is tangent to circle OO at AA, and PBCPBC is a secant where PB=4PB = 4 and PC=9PC = 9. Find PAPA.

Example 13

hard
Two circles with centers 44 apart have radii 11 and 22. Find the length of a common external tangent segment between the tangent points.

Example 14

hard
Two circles with centers 1010 apart and radii 33 and 55. Find the length of a common internal tangent.

Example 15

hard
From external point PP, two tangents to circle OO make an angle of ∠APB=90°\angle APB = 90° at PP. If the radius is r=6r = 6, find OPOP.

Example 16

hard
A circle of radius rr is inscribed in a square of side 1010. The circle is tangent to all four sides. Find rr.

Example 17

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

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

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