# TOPIC 6: CIRCLES ~ MATHEMATICS FORM 3

**TOPIC 6: CIRCLES ~ MATHEMATICS FORM 3**

**Definition of Terms**

Circle, Chord, Radius, Diameter, Circumference, Arc, Sector, Centre and Segment of a Circle

Define circle, chord, radius, diameter, circumference, arc, sector, centre and segment of a circle

A

**circle:**is the locus or the set of all points equidistant from a fixed point called the center.**Arc:**a curved line that is part of the circumference of a circle

**Chord:**a line segment within a circle that touches 2 points on the circle.

**Circumference:**The distance around the circle.

**Diameter:**The longest distance from one end of a circle to the other.

**Origin:**the center of the circle

**Pi(π):**A number, 3.141592…, equal to (the circumference) / (the diameter) of any circle.

**Radius:**distance from center of circle to any point on it.

**Sector:**is like a slice of pie (a circle wedge).

**Tangent of circle:**a line perpendicular to the radius that touches ONLY one point on the circle.

**NB:**

**Diameter = 2 x radius of circle**

Circumference of Circle =

**PI x diameter**= 2 PI x radius

**Tangent Properties**

A Tangent to a Circle

Describe a tangent to a circle

is a line which touches a circle. The point where the line touches the

circle is called the point of contact. A tangent is perpendicular to the

radius at the point of contact.

Tangent Properties of a Circle

Identify tangent properties of a circle

tangent to a circle is perpendicular to the radius at the point of

tangency. A common tangent is a line that is a tangent to each of two

circles. A common external tangent does not intersect the segment that

joins the centers of the circles. A common internal tangent intersects

the segment that joins the centers of the circles.

Tangent Theorems

Prove tangent theorems

Theorem 1

If

two chords intersect in a circle, the product of the lengths of the

segments of one chord equal the product of the segments of the other.

two chords intersect in a circle, the product of the lengths of the

segments of one chord equal the product of the segments of the other.

**Intersecting Chords Rule:**(segment piece)×(segment piece) =(segment piece)×(segment piece)

**Theorem Proof:**

**Theorem 2:**

**Secant-Secant Rule:**(whole secant)×(external part) =(whole secant)×(external part)

Theorems Relating to Tangent to a Circle in Solving Problems

Apply theorems relating to tangent to a circle in solving problems

Example 7

Two

common tangents to a circle form a minor arc with a central angle of

140 degrees. Find the angle formed between the tangents.

common tangents to a circle form a minor arc with a central angle of

140 degrees. Find the angle formed between the tangents.

**Solution**

Two tangents and two radii form a figure with 360°. If y is the angle formed between the tangents then y + 2(90) + 140° = 360°

y = 40°.

The angle formed between tangents is 40 degrees.