### FORM THREE MATHEMTICS STUDY NOTES TOPIC 6:**CIRCLES**

**Definition of Terms**

A Tangent to a Circle

Describe a tangent to a circle

Tangent Properties of a Circle

Identify tangent properties of a circle

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.

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

**Theorem Proof:**

**Theorem 2:**

If two secant segments are drawn to a circle from the same external point, the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part.

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

**Theorem 3:**

If a secant segment and tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.

**Secant-Tangent Rule:**(whole secant)×(external part) =(tangent)2

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.

**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.