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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