For the two circles in Figures a, b, and c, find the

43. For the drawing in Exercise 42, suppose that the larger

gear has 20 teeth and the smaller gear has 10 the gear ratio is 2:1. If the smaller gear rotates through an angle of 90°, through what angle measure does the larger gear rotate? In Exercises 44 to 47, prove the stated theorem.

44. If a line is drawn through the center of a circle

perpendicular to a chord, then it bisects the chord and its minor arc. See Figure 6.37. NOTE: The major arc is also bisected by the line.

45. If a line is drawn through the center of a circle to the

midpoint of a chord other than a diameter, then it is perpendicular to the chord. See Figure 6.38.

46. If two secant segments are drawn to a circle from an

external point, then the products of the lengths of each secant with its external segment are equal. See Figure 6.51.

47. If a tangent segment and a secant segment are drawn to a

circle from an external point, then the square of the length of the tangent equals the product of the length of the secant with the length of its external segment. See Figure 6.52. Construction of Tangents to a Circle Inequalities in the Circle Some Constructions and Inequalities for the Circle

6.4

KEY CONCEPTS In Section 6.3, we proved that the radius drawn to a tangent at the point of contact is perpendicular to the tangent at that point. We now show, by using an indirect proof, that the converse of that theorem is also true. Recall that there is only one line perpendicu- lar to a given line at a point on that line. The line that is perpendicular to the radius of a circle at its endpoint on the circle is a tangent to the circle. THEOREM 6.4.1 GIVEN: In Figure 6.54a, with radius PROVE: is a tangent to at point T PROOF: Suppose that is not a tangent to at T. Then the tangent call it can be drawn at T, the point of tangency. [See Figure 6.54b.] Now is the radius to tangent at T, and because a radius drawn to a tangent at the point of contact of the tangent is perpendicular to the tangent, . But by hypothesis. Thus, two lines are perpendicular to at point T, contradicting the fact that there is only one line perpendicular to a line at a point on the line. Therefore, must be the tangent to at point T. }O Í QT OT OT ⬜ Í QT OT ⬜ Í RT Í RT OT Í RT }O Í QT }O Í QT Í QT ⬜ OT OT }O O T Q a b R O T Q Figure 6.54