Master

Tangent to a Circle

Master tangent to a circle with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Tangent to a Circle

Study Guide

Key Definition

A tangent to a circle is a straight line that touches the circle at only one point. This point is called the point of tangency. The tangent is perpendicular to the radius at the point of tangency.

Important Notes

Point P is the point of tangency and T is an external point; the line $\overline{PT}$ is a tangent to circle $O$
The radius $\overline{OP}$ is perpendicular to the tangent $\overline{PT}$
Use the Pythagorean Theorem to find distances in right triangles formed by tangents
$\angle OPT = 90^\circ$
The length of $\overline{OT}$ is the hypotenuse in $\triangle OPT$

Mathematical Notation

$\angle$ means angle

$\triangle$ means triangle

$\perp$ means perpendicular

$\overline{AB}$ means line segment AB

$+$ represents addition

Remember to use proper notation when solving problems

Why It Works

The tangent is perpendicular to the radius at the point of tangency, allowing us to use the Pythagorean Theorem with $(OP)^2 + (PT)^2 = (OT)^2$ to find unknown lengths.

Remember

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

Quick Reference

Pythagorean Theorem:$a^2 + b^2 = c^2$

Understanding Tangent to a Circle

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A tangent is a line that touches a circle at one point forming a right angle with the radius.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $OP = 3$ units and $PT = 4$ units, what is the length of $OT$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is built tangent to a circular garden with a radius of $6$ meters. How far is the base of the ramp from the circle's center if the tangent segment from the base of the ramp to the point of tangency is $8$ meters?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a circle and a tangent, prove geometrically why the tangent is perpendicular to the radius at the point of tangency.

Challenge Quiz

Single Choice Quiz

Advanced

If $OP = 9$ and the tangent $PT$ is $12$, find the length of $OT$.

Recap

Watch & Learn

Review key concepts and takeaways