Concurrency of Perpendicular

Transcription

1 re-a Geometry Date 5.2 Notes isectors of a triangle (pp ) age 1 of 5 I can define a concurrent lines and identify the point of concurrency. I can state the point of concurrency of perpendicular bisectors and angle bisectors of a triangle. I can state where the point of concurrency of perpendicular bisectors will be for types of triangles. I can state and apply the concurrency of angle bisectors and perpendicular bisectors. I can solve problems and perform proofs with perpendicular bisectors of triangles. 5.5 Concurrency of erpendicular isectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. A A = = C C 5.6 Concurrency of Angle isectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. D = E = D A E C

2 circumcenter is at. The circumcenter of a triangle has a special property, as described in Theorem 5.5. You will use coordinate geometry to illustrate this theorem in Exercises A proof appears on page 835. re-a Geometry Date 5.2 Notes isectors of a triangle (pp ) age 2 of 5 When three or more lines (or rays or segments) intersect in the same point, they are called concurrent 5.5 Concurrency lines (or rays of erpendicular or segments). The point of intersection of isectors of a Triangle the lines is called the point of concurrency. The perpendicular bisectors of a triangle The three perpendicular bisectors of a triangle are concurrent. The point of intersect at a point that is equidistant from A concurrency the can vertices be of inside the triangle. the triangle, on the triangle, or outside the triangle. A = = C The diagram for Theorem 5.5 shows that the circumcenter is the center of the circle that passes through the vertices of the triangle. The circle is circumscribed about AC. Thus, the radius of this circle is the distance from the center to any of the vertices. C acute EXAMLE triangle 1 Using erpendicular right isectors triangle obtuse triangle ACILITIES LANNING A company plans to build a distribution center that is convenient to three of its major clients. The planners start by roughly locating the three clients on a sketch and finding the circumcenter of the triangle formed. REAL LIE a. Explain why using the circumcenter as the location of a distribution center would be convenient for all the clients. b. Make a sketch of the triangle formed by the clients. Locate the circumcenter of the triangle. Tell what segments are congruent. E G HEL EWORK HEL t our Web site ugallittell.com amples. a. ecause the circumcenter is equidistant from the three vertices, each client would be equally close to the distribution center. b. Label the vertices of the triangle as E,, and G. Draw the perpendicular bisectors. Label their intersection as D. y Theorem 5.5, DE = D = DG. E D G 5.2 isectors of a Triangle 273

3 re-a Geometry Date 5.2 Notes isectors of a triangle (pp ) age 3 of 5 Three people need to decide on a location to hold a monthly meeting. They will all be coming from different places in the city, and they want to make the meeting location the same distance from each person. 1. Explain why using the circumcenter as the location for the meeting would be the fairest for all. 2. The triangle with the circumcenter constructed is shown. Label the circumcenter. State which segments (that aren t labeled) are congruent. 3. Which point is the best spot for placing a sprinkler to water the plants located at point X, Y, and Z? 4. If the sprinkler covers a circular region with a radius of 15 in., will the water reach all three plants?

4 re-a Geometry Date 5.2 Notes isectors of a triangle (pp ) age 4 of 5 EXAMLE 2 Using Angle isectors The angle bisectors of MN meet at point L. a. What segments are congruent? b. ind LQ and LR. a. y Theorem 5.6, the three angle bisectors of a triangle intersect at a point that is equidistant from Æ Æ Æ the sides of the triangle. So, LR LQ LS. STUDENT HEL b. Use the ythagorean Theorem to find LQ in LQM. (LQ) 2 + (MQ) 2 = (LM) 2 (LQ) = 17 2 (LQ) = 289 (LQ) 2 = 64 LQ = 8 Substitute. Multiply. Logical Reasoning Look ack or help with the ythagorean Theorem, see p. 20. Subtract 225 from each side. ind the positive square root. Æ Æ So, LQ = 8 units. ecause LR LQ, LR = 8 units. M S 17 Using Angle isectors Chapter 5 roperties of Triangles The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. The diagram for Theorem 5.6 shows that the inc that touches each side of the triangle once. The c Thus, the radius of this circle is the distance from EXAMLE 2 The angle bisectors of MN meet at point L. a. What segments are congruent? q b. ind LQ and LR. D = E = a. y Theorem 5.6, the three angle bisectors of triangle intersect at a point that is equidistan Æ Æ Æ the sides of the triangle. So, LR LQ LS b. Use the ythagorean Theorem to find LQ in L (LQ) 2 + (MQ) 2 = (LM) 2 (LQ) = 17 2 (LQ) = 289 (LQ) 2 = 64 LQ = 8 N R Substitute. Multiply. Subtract 225 f ind the posit Æ Æ So, LQ = 8 units. ecause LR LQ, LR

5 re-a Geometry Date 5.2 Notes isectors of a triangle (pp ) age 5 of 5 The angle bisectors of meet at. 5. What segments are congruent? 6. ind T & V. 7. The angle bisectors of meet at L. ind AL & L. 8. What name is given to 3 or more lines that intersect at one point?