A surveyor sees a circular

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 GIVEN: In Figure 6.39a, is the perpendicular bisector of chord in PROVE: contains point O PROOF BY INDIRECT Suppose that O is not on . Draw and radii METHOD: and . [See Figure 6.39b.] Because is the perpendicular bisector of , R must be the midpoint of ; then . Also, all radii of a are . With by Identity, we have by SSS. Now by CPCTC. It follows that because these line segments meet to form congruent adjacent angles. Then is the perpendicular bisector of . But is also the perpendicular bisector of , which contradicts the uniqueness of the perpendicular bisector of a segment. Thus, the supposition must be false, and it follows that center O is on , the perpendicular bisector of chord . TV Í QR TV Í QR TV OR OR ⬜ TV ∠ORT ⬵ ∠ORV 䉭ORT ⬵ 䉭ORV OR ⬵ OR ⬵ } OT ⬵ OV TR ⬵ RV TV TV Í QR OV OT OR Í QR Í QR }O TV Í QR EXAMPLE 1 GIVEN: In Figure 6.40, has a radius of length 5 at B and FIND: CD Solution Draw radius . By the Pythagorean Theorem, According to Theorem 6.3.1, we know that ; then it follows that . CD = 2 4 = 8 CD = 2 BC BC = 4 BC 2 = 16 25 = 9 + BC 2 5 2 = 3 2 + BC 2 OC 2 = OB 2 + BC 2 OC OB = 3 OE ⬜ CD }O O C D B E Figure 6.40 Figure 6.41 쮿 CIRCLES THAT ARE TANGENT In this section, we assume that two circles are coplanar. Although concentric circles do not intersect, they do share a common center. For the concentric circles shown in Figure 6.41, the tangent of the smaller circle is a chord of the larger circle. If two circles touch at one point, they are tangent circles. In Figure 6.42, circles P and Q are internally tangent, whereas circles O and R are externally tangent. Q a P b O R Figure 6.42 Exs. 1–4 쮿