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