Properties of Tangents to a Circle

Understand tangent lines and their properties. Grade 9 Alberta mathematics circle geometry.

Lesson 8.1 of Circle Geometry in Grade 9 Math — Alberta curriculum lessons.

What you'll learn in this lesson

• Key Definitions. Learn important circle geometry terms.
• Understanding Tangents and Angle Relationships. Explore what tangents are, how they interact with the radius, and how to find missing angles using tangent–radius properties.
• Using the Pythagorean Theorem in Circle Problems. Apply the Pythagorean Theorem to solve for unknown sides and distances involving tangents and radii.

Inside the lesson — a free preview

What You'll Learn

Before learning about tangents, let's review the important terms used in circle geometry.

These words and diagrams will come up throughout Unit 8.

Radius

Radius. The distance or line segment from the center of the circle to any point on the circle.

Diameter

Diameter.

Chord

Chord.

Arc and Minor Arc

Arc. A part of the circumference (outer edge) of a circle.

Minor Arc.

Tangent

Tangent.

Point of Tangency

Point of Tangency. The exact point where a tangent touches or intersects the circle.

Central Angle

Central Angle. An angle whose vertex is at the center of the circle and whose arms are radii of the circle.

Inscribed Angle

Inscribed Angle. An angle whose vertex and arms are on the circle itself.

Example: ∠PQR has its vertex Q on the circle, and both arms lie along the circumference.

What Is a Tangent?

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

It does not enter or cut across the circle — it only touches the outside edge once.

That point where it touches is called the point of tangency.

Tangent–Radius Property

A tangent to a circle is perpendicular (90°) to the radius at the point of tangency.

This means the radius drawn to the point where the tangent touches the circle always makes a right angle with the tangent line.

Worked example: Example 1

Question. Point O is the center of a circle and AB is tangent to the circle at A.

In △OAB, ∠AOB = 55°.

Determine the measure of ∠OBA.

Step 1. Since OA is a radius and AB is a tangent,

∠OAB = 90°.

Step 2. Add the two known angles:

90° + 55° = 145°.

Step 3. The sum of all angles in a triangle is 180°.

180° − 145° = 35°.

✅ Final Answer. ∠OBA = 35°

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