Given: and are Given: Given: Tangent to at

37. The larger circle is inscribed in a square with sides of

length 4 cm. The smaller circle is tangent to the larger circle and to two sides of the square as shown. Find the radius of the smaller circle. Q R V P S T S R V T Q

38. In , .

Also, . Find . m ∠VRS m ∠P = 23° QS = 2PT }R In Exercises 39 to 47, provide a paragraph proof. Be sure to provide a drawing, Given, and Prove where needed.

39. If two parallel lines intersect a circle, then the intercepted

arcs between these lines are congruent. HINT: See Figure 6.36. Draw chord

40. The line joining the centers of two circles that intersect at

two points is the perpendicular bisector of the common chord. AD.

41. If a trapezoid is inscribed in a circle, then it is an isosceles

trapezoid.

42. If a parallelogram is inscribed in a circle, then it is a

rectangle.

43. If one side of an inscribed triangle is a diameter, then the

triangle is a right triangle.

44. Prove Case 2 of Corollary 6.2.4: The measure of an angle

formed by a tangent and a chord drawn to the point of tangency is one-half the measure of the intercepted arc. See Figure 6.29.

45. Prove Case 3 of Corollary

6.2.4. See Figure 6.29.

46. Given: with P in its

exterior; O-Y-P Prove: OP 7 OY }O O P X Y D

47. Given:

Quadrilateral RSTV inscribed in Prove: m ∠R + m∠T = m∠V + m∠S } Q Tangent Circles Internally Tangent Circles Externally Tangent Circles Line of Centers Common Tangent Common External Tangents Common Internal Tangents Line and Segment Relationships in the Circle 6.3 KEY CONCEPTS In this section, we consider further line and line segment relationships in the circle. Because some statements such as Theorems 6.3.1–6.3.3 are so similar in wording, the student is strongly encouraged to make drawings and then compare the information that is given in each theorem to the conclusion of that theorem. If a line is drawn through the center of a circle perpendicular to a chord, then it bisects the chord and its arc. THEOREM 6.3.1 NOTE: Note that the term arc generally refers to the minor arc, even though the major arc is also bisected. GIVEN: in circle A See Figure 6.37. PROVE: and The proof is left as an exercise for the student. HINT: Draw and . Even though the Prove statement does not match the conclusion of Theorem 6.3.1, we know that is bisected by if and that is bisected by if . CE ¬ ⬵ ED ¬ Í AE CD ¬ CB ⬵ BD Í AB CD AD AC CE ¬ ⬵ ED ¬ CB ⬵ BD Í AB ⬜ chord CD A E C D B Figure 6.37 O R S M Figure 6.38 T V R a Q O O T V R b Q Figure 6.39 If a line through the center of a circle bisects a chord other than a diameter, then it is perpendicular to the chord. THEOREM 6.3.2 GIVEN: Circle O; is the bisector of chord See Figure 6.38. PROVE: The proof is left as an exercise for the student. HINT: Draw radii and . Figure 6.39a illustrates the following theorem. However, Figure 6.39b is used in the proof. OS OR Í OM ⬜ RS RS Í OM The perpendicular bisector of a chord contains the center of the circle. THEOREM 6.3.3