Properties of Chords in a Circle

Learn about chords and their geometric properties. Grade 9 Alberta mathematics.

Lesson 8.2 of Circle Geometry in Grade 9 Math: Alberta curriculum lessons.

What you'll learn in this lesson

• Understanding Chords. Learn what a chord is, and key theory
• Examples. Apply the chord and perpendicular bisector properties to solve geometry problems involving circles.

Inside the lesson: a free preview

What is a Chord?

A chord is a line segment that joins two points on a circle.

Every circle has many possible chords.

The diameter is a special type of chord. It passes through the center of the circle and is the longest possible chord that can be drawn.

Perpendicular to Chord Property

"In any circle with center O and chord AB, then:"

This property links geometry and symmetry in circles. It shows that the center of a circle lies on the perpendicular bisector of every chord.

Visual Explanation.

Key Facts We'll Use

Perpendicular to Chord Property. A line from the center that is perpendicular to a chord bisects the chord.

Pythagorean Theorem. When a radius is perpendicular to a chord, use:

a² + b² = c²

where c is the hypotenuse (longest side, opposite 90°).

Worked example: Example 1

Question. Find the length of chord AB

Given.

Step 1: Identify the triangle. We can see a right triangle formed by O, C, and B.

Step 2: Use the Pythagorean Theorem. Formula:

a² + b² = c²

Substitute the values:

6² + BC² = 10²

36 + BC² = 100

BC² = 100 − 36

BC² = 64

BC = √64 = 8 cm

Step 3: Use the perpendicular bisector property. Since OC ⊥ AB, point C is the midpoint of AB.

So AC = BC = 8 cm.

Step 4: Find the full chord. AB = 2 × 8 = 16 cm

✅ Final Answer. AB = 16 cm

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