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Sect. 12-2 Properties of Tangents

Sect. 12-2 Properties 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?

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Sect. 12-2 Properties of Tangents

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  1. Sect. 12-2Properties of Tangents Geometry Honors

  2. 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.

  3. 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.

  4. 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.

  5. 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 .

  6. Example • is tangent to circle C at B. Find the length of a radius of circle C. • Since is a tangent to circleC at B, is a right triangle with hypotenuse .

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

  8. Example Continued • ABCE is a rectangle. is a right triangle. • Use Pythagorean theorem to solve for AD.

  9. 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.

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

  11. 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.

  12. 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 .

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

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

  15. 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

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