Given: with and TEST 534

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 CONSTRUCTIONS OF TANGENTS TO CIRCLES P a X b P X Z Y P c X Z Y W Figure 6.55 Construction 8 To construct a tangent to a circle at a point on the circle. PLAN: The strategy used in Construction 8 is based on Theorem 6.4.1. For Figure 6.55a, we will draw a radius extended beyond the circle. At the point on the circle point X in Figure 6.55, we construct the line perpendicular to The constructed line [ in Figure 6.55c] is tangent to circle P at point X. GIVEN: with point X on the circle [See Figure 6.55a.] CONSTRUCT: A tangent to at point X }P Í XW }P Í WX PX . CONSTRUCTION: Figure 6.55a: Consider and point X on . Figure 6.55b: Draw radius and extend it to form . Using X as the center and any radius length less than XP , draw two arcs to intersect at points Y and Z. Figure 6.55c: Complete the construction of the perpendicular line to at point X. From Y and Z, mark arcs with equal radii of length greater than XY. Calling the point of intersection W, draw , the desired tangent to at point X. }P Í XW PX PX PX PX }P }P EXAMPLE 1 Make a drawing so that points A, B, C, and D are on in that order. If tangents are constructed at points A, B, C, and D, what type of quadrilateral will be formed by the tangent segments if a and ? b all arcs , , , and are congruent? Solution a A rhombus all sides are congruent b A square all four are right ; all sides We now consider a more difficult construction. ⬵ ∠s ∠s DA ¬ CD ¬ BC ¬ AB ¬ mBC ¬ = mAD ¬ m AB ¬ = mCD ¬ }O 쮿