Sect. 12-2Properties of Tangents

Geometry Honors

What and Why

• What?– Find the relationship between a radius and a

tangent, and between two tangents drawn from the same point.

– Circumscribe a circle• Why?– To use tangents to circles in real-world situations,

such as working in a machine shop.

Tangents to Circles

• We have looked at the tangent of an angle in a right triangle. Now we will look at the properties of a tangent to a circle.

• A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point.

• The point where a circle and a tangent intersect is the point of tangency.

Continued

• is a tangent ray and is a tangent segment. The word tangent may refer to a tangent line, tangent ray, or a tangent segment.

Theorem 12-2

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

• If is tangent to circle N at A, then .

Example

• is tangent to circle C at B. Find the length of a radius of circle C.

• Since is a tangent to circle C at B, is a right triangle with hypotenuse .

•

Example

• Machine Shop – A belt fits tightly around two circular pulleys. Find the distance between the centers of the pulleys.

Example Continued

• ABCE is a rectangle. is a right triangle.

• Use Pythagorean theorem to solve for AD.

Theorem 12-3Converse of Theorem 12-2

• If a line in the same plane as a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

• If , then is tangent to circle O.

Example

• In the diagram, is tangent to circle N at L?

Circumscribing Circles

• In the figure, the sides of the triangle are tangent to the circle. The triangle is circumscribed about the circle. The circle is inscribed in the triangle.

Theorem 12-4

• Two segments tangent to a circle from a point outside the circle are congruent.

• If and are tangent to circle O at A and B respectively, then .

Example

• Circle O is inscribed in . Find the perimeter of .

Example

• The diagram represents a chain drive system on a bicycle. Is BC = GF?

Solution

• Yes. Extend and to intersect in point H.

• By Theorem 11-3, HC = HF, or HB + BC = HG + GF. By Theorem 11-3 again, HB = HG, so by the Subtraction Property of Equality, BC = GF