**Chapter 10.1 Tangents to Circles**

**I) Vocabulary:**

**A) Circle**- is the set of all pints in a plane that are equidistant from a given point, called the center of the circle.

**B) Radius**- the distance from the center to a point on the circle. Two circles are congruent if they have the same radius.

**C) Diameter**- the distance across the circle, through its center. The diameter is twice the radius.

**D) Chord**- is a segment whose endpoints are points on the circle.

**E) Diameter**- is a chord that passes through the center of the circle.

**F) Secant**- is a line that intersects a circle in two points.

**G) Tangent**- is a line in the plane of a circle that intersects the circle in exactly one point.

**H) Tangent Circles**- coplanar circles that intersect in one point.

**I) Concentric**- coplanar circles that have a common center.

**J) Common tangents**- a line or segment that is tangent to two coplanar circles.

**1) Common internal tangent**- intersects the segment that joins the centers of the two circles.

**2) Common external tangent**- does not intersect the segment that joins the centers of the two circles.

**K) Interior of a circle**- consists of the points that are inside the circle.

**L) Exterior of a circle**- consists of the points that are outside the circle.

**M) Point of tangency**- the point at which a tangent line intersects the circle to which it is tangent.

**N) Theorems:**

1) If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

**Example**: Given line AB is tangent to circle C with point A on circle C, AC is a radius of 5 units, AB is 12 units, what is the length of line segment CB?

Since line AB is a tangent line, we know that AC and AB are perpendicular so angle CAB is a right angle so triangle CAB is a right triangle. Therefore we can use pythagorean theorem to find CB.

5

^{2}+ 12

^{2}= (CB)

^{2}

25 + 144 = (CB)

^{2}

169 = (CB)

^{2}

13 = CB

2) In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

**Example:**Given radius DE = 11 units on circle D, there is an external point F, line segment DF = 61 and line segment EF = 60, is line EF a tangent line to circle D?

11

^{2}+ 60

^{2}= 121 + 3600 = 3721

61

^{2}= 3721

Therefore, since the two smaller sides squared are equal to the longer side squared, this is the converse of pythagorean theorem so we know that angle DEF is a right angle, so therefore line EF is perpendicular to line segment DE so therefore EF is a tangent line to circle D.

3) If two segments from the same exterior point are tangent to a circle, then they are congruent.

**Example:**Given tangent SR = 2x + 7 and tangent SB = 5x - 8, where both tangents intersect at point S off of circle C with points R and B are on circle C, solve for x.

2x + 7 = 5x - 8

7 = 3x - 8

15 = 3x

5 = x