#### CIRCLES

**Definition of Terms**

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:**

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.

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.

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.

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.