Properties of Tangents to a Circle
2026-02-28 01:02 Diff
https://brightchamps.com/en-us/math/geometry/properties-of-tangents-to-a-circle

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Last updated on September 10, 2025

Tangents to a circle have several unique properties that are essential in solving geometric problems related to circles. These properties help students analyze and solve problems involving tangents, angles, and distances. The properties include: a tangent to a circle is perpendicular to the radius at the point of tangency, and tangents drawn from a common external point are equal in length. Understanding these properties assists students in solving problems related to circles effectively. Now let us learn more about the properties of tangents to a circle.

What are the Properties of Tangents to a Circle?

The properties of tangents to a circle are fundamental and help students understand and work with this geometric concept. These properties are derived from the principles of geometry. There are several properties of tangents to a circle, and some of them are mentioned below:

Property 1: Perpendicularity

A tangent to a circle is perpendicular to the radius at the point of tangency.

Property 2: Equal Tangents

Tangents drawn from a common external point to a circle are equal in length.

Property 3: Angle Between Tangents

The angle between two tangents drawn from a common external point is bisected by the line segment joining the center of the circle to the external point.

Property 4: Tangent-Secant

The square of the length of a tangent drawn from an external point to a circle is equal to the product of the lengths of the secant segment and its external segment from the same point.

Tips and Tricks for Properties of Tangents to a Circle

Students often find the properties of tangents to a circle confusing. To avoid such confusion, we can follow these tips and tricks:

Perpendicularity to the Radius: Students should remember that a tangent to a circle is always perpendicular to the radius at the point of tangency. They can verify this by drawing a line from the circle's center to the tangent point and observing the right angle formed.

Equal Tangents: Students should recall that tangents drawn from a common external point are equal in length. To verify this, students can measure the tangents from the same external point to different points of tangency on the circle.

Tangent-Secant Relationship: Students should remember that the square of the tangent length equals the product of the whole secant segment and its external segment from the same external point.

Confusing Tangents with Secants

Students should remember that a tangent touches the circle at one point, whereas a secant intersects the circle at two points.

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Problem 1

Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(10² - 6²) = √(100 - 36) = √64 = 8 cm.

Okay, lets begin

Two tangents PA and PB are drawn from an external point P to a circle with center O. If PA = 15 cm, what is the length of PB?

Well explained 👍

Problem 2

Since tangents drawn from a common external point are equal, PA = PB. Thus, PB = 15 cm.

Okay, lets begin

In a circle, a tangent at point A meets a line segment OP where O is the center of the circle. If OA = 5 cm and OP = 13 cm, find the length of PA.

Well explained 👍

Problem 3

Using the Pythagorean theorem, since OP is the hypotenuse and OA is perpendicular, PA = √(OP² - OA²) = √(13² - 5²) = √(169 - 25) = √144 = 12 cm.

Okay, lets begin

If a tangent from a point P to a circle is 9 cm long and the distance from P to the center of the circle is 15 cm, what is the radius of the circle?

Explanation

Radius = 12 cm.

Well explained 👍

Problem 4

Using the Pythagorean theorem, since OP is the hypotenuse and PA is perpendicular, Radius = √(OP² - PA²) = √(15² - 9²) = √(225 - 81) = √144 = 12 cm.

Okay, lets begin

A circle has center O, and two tangents PA and PB are drawn from point P such that PA = 6 cm. If angle APB = 60 degrees, what is angle AOB?

Explanation

Angle AOB = 120 degrees.

Well explained 👍

A tangent to a circle is a straight line that touches the circle at exactly one point.

1.How many tangents can be drawn from a point outside a circle?

Two tangents can be drawn from a point outside a circle.

2.Are tangents from the same external point equal?

Yes, tangents drawn from the same external point to a circle are equal in length.

3.How do you find the length of a tangent?

To find the length of a tangent, you can use the Pythagorean theorem if the radius and the distance from the external point to the center of the circle are known.

4.Can a tangent pass through the circle?

No, a tangent can only touch the circle at one point and cannot pass through it.

Common Mistakes and How to Avoid Them in Properties of Tangents to a Circle

Students tend to get confused when understanding the properties of tangents to a circle, and they often make mistakes while solving problems related to these properties. Here are some common mistakes and solutions to avoid them.

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.