Two congruent intersecting circles B and D not shown

29. Given: Quadrilateral

ABCD is circumscribed about Prove:

30. Given: in

Prove:

31. Does it follow from Exercise 30

that is also congruent to ? What can you conclude about and in the drawing? What can you conclude about and ?

32. In not shown,

is a diameter and T is the midpoint of semicircle . What is the value of the ratio ? The ratio ?

33. The cylindrical brush

on a vacuum cleaner is powered by an electric motor. In the figure, the drive shaft is at point D. If find the measure of the angle formed by the drive belt at point D; that is, find .

34. The drive mechanism on a treadmill is powered by an

electric motor. In the figure, find if is 36° larger than

35. Given: Tangents , , and to at

points M, N, and P, respectively , , Find: AM, PC, and BN AC = 12 BC = 16 AB = 14 }O AC BC AB mAC ¬. m ABC ២ m ∠D m ∠D mAC ¬ = 160°, RT RO RT RS RTS ២ RS }O EB DE CE AE 䉭CBE 䉭ADE 䉭ABD ⬵ 䉭CDB }P AB ⬵ DC DA + BC AB + CD = }O C A D B E P Exercises 30, 31 C A D B C A D B Exercises 33, 34 C D B A O N M C A B P O B A O D P Exercises 38, 39 Exercises 42, 43

36. Given: is inscribed in

isosceles right The perimeter of is Find: TM 8 + 4 22 䉭RST 䉭RST }Q N M R S P Q T

37. Given: is an external

tangent to and at points A and B ; radii lengths for and are 4 and 9, respectively Find: AB HINT: The line of centers contains point C, the point at which and are tangent.

38. The center of a circle of radius 3 inches is at a distance

of 20 inches from the center of a circle of radius 9 inches. What is the exact length of common internal tangent HINT: Use similar triangles to find OD and DP. Then apply the Pythagorean Theorem twice. AB ? }Q }O OQ }Q }O }Q }O AB B A O Q C

39. The center of a circle of radius 2 inches is at a distance

of 10 inches from the center of a circle of radius length 3 inches. To the nearest tenth of an inch, what is the approximate length of a common internal tangent? Use the hint provided in Exercise 38.

40. Circles O, P, and Q are

tangent as shown at points X, Y, and Z. Being as specific as possible, explain what type of triangle is if: a , , b , ,

41. Circles O, P, and Q are tangent as shown at points X, Y,

and Z. Being as specific as possible, explain what type of triangle is if: a , , b , ,

42. If the larger gear has 30 teeth and the smaller gear has

18, then the gear ratio larger to smaller is 5:3. When the larger gear rotates through an angle of 60°, through what angle measure does the smaller gear rotate? QZ = 2 PY = 2 OX = 2 QZ = 1 PY = 4 OX = 3 䉭PQO QZ = 2 PY = 3 OX = 2 QZ = 1 PY = 3 OX = 2 䉭PQO O Q P X Y Z Exercises 40, 41

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