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5.3 Bisectors of a Triangle

5.3 Bisectors of a Triangle. Geometry. Objectives. Objectives: You will use properties of angle bisectors of a triangle. DFA: #18 – p.305 HW pp.304-307 (2-32 even). Using Perpendicular Bisectors of a Triangle.

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5.3 Bisectors of a Triangle

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  1. 5.3 Bisectors of aTriangle Geometry

  2. Objectives • Objectives: • You will use properties of angle bisectors of a triangle. • DFA: • #18 – p.305 • HW • pp.304-307 (2-32 even)

  3. Using Perpendicular Bisectors of a Triangle • In Lesson 5.1, you studied the properties of perpendicular bisectors of segments and angle bisectors. In this lesson, you will study the special cases in which segments and angles being bisected are parts of a triangle.

  4. Perpendicular Bisector of a Triangle • A perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side. Perpendicular Bisector

  5. Class Activity – p.300

  6. Notes: • When three or more concurrent lines (or rays or segments) intersect in the same point, then they are called concurrent lines (or rays or segments). The point of intersection of the lines is called the point of concurrency.

  7. About concurrency • The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle. 90° Angle-Right Triangle

  8. About concurrency • The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle. Acute Angle-Acute Scalene Triangle

  9. About concurrency • The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle. Obtuse Angle-Obtuse Scalene Triangle

  10. Notes: • The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenterof the triangle. In each triangle, the circumcenter is at point P. 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 29-31. A proof appears for your edification on pg. 835.

  11. Theorem 5.6 Concurrency of Perpendicular Bisectors of a Triangle • The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. • BA = BD = BC

  12. What about the circle? • The diagram for Theorem 5.6 shows that a circumcenter is the center of the circle that passes through the vertices of the triangle. The circle is circumscribed about ∆ACD. Thus the radius of this circle is the distance from the center to any of the vertices.

  13. Ex. 2: Using perpendicular Bisectors—pg. 303 • FACILITIES PLANNING. A city planner plans to build a fire station that is equal distances from the elementary, middle & high schools. Where should she locate the fire station based on the diagram?

  14. Using angle bisectors of a triangle • An angle bisector of a triangle is a bisector of an angle of the triangle. The three angle bisectors are concurrent. The point of concurrency of the angle bisectors is called the incenter of the triangle, and it always lies inside the triangle. The incenter has a special property that is described in Theorem 5.7. Exercise 24 asks you to write a proof of this theorem.

  15. Theorem 5.7 Concurrency of Angle Bisectors of a Triangle • The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. • PD = PE = PF

  16. Notes: • The diagram for Theorem 5.7 shows that the incenter is the center of the circle that touches each side of the triangle once. The circle is inscribed within ∆ABC. Thus the radius of this circle is the distance from the center to any of the sides.

  17. Ex. 2 Using Angle Bisectors • The angle bisectors of ∆MNP meet at point L. • What segments are congruent? Find LQ and LR. • ML = 17 • MQ = 15

  18. By Theorem 5.7, the three angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. So, LR  LQ  LS

  19. b. Use the Pythagorean Theorem to find LQ in ∆LQM a2 + b2 = c2 (LQ)2 + (MQ)2 = (LM)2 Substitute (LQ)2 + (15)2 = (17)2 Substitute values (LQ)2 + (225) = (289) Multiply (LQ)2 = (64) Subtract 225 from each side. LQ = 8 Find the positive square root ►So, LQ = 8 units. Because LR LQ, LR = 8 units

  20. #22 Developing Proof. Complete the proof of Theorem 5.7 the Concurrency of Angle Bisectors Given►∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCA Prove►The angle bisectors intersect at a point that is equidistant from AB, BC, and CA

  21. Given►∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCAProve►The angle bisectors intersect at a point that is equidistant from AB, BC,and CA Reasons: • Given Statements: • ∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCA • ______ = DG • DE = DF • DF = DG • D is on the ______ of C. • ________

  22. Given►∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCAProve►The angle bisectors intersect at a point that is equidistant from AB, BC,and CA Reasons: • Given • AD bisects BAC, so D is equidistant from the sides of BAC Statements: • ∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCA • DE = DG • DE = DF • DF = DG • D is on the ______ of C. • ________

  23. Given►∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCAProve►The angle bisectors intersect at a point that is equidistant from AB, BC,and CA Reasons: • Given • AD bisects BAC, so D is___ from the sides of BAC • BD bisects ABC, so D is equidistant from the sides of ABC. Statements: • ∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCA • DE = DG • DE = DF • DF = DG • D is on the ______ of C. • ________

  24. Given►∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCAProve►The angle bisectors intersect at a point that is equidistant from AB, BC,and CA Reasons: • Given • AD bisects BAC, so D is___ from the sides of BAC • BD bisects ABC, so D is equidistant from the sides of ABC. • Trans. Prop of Equality Statements: • ∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCA • DE = DG • DE = DF • DF = DG • D is on the ______ of C. • ________

  25. Given►∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCAProve►The angle bisectors intersect at a point that is equidistant from AB, BC,and CA Reasons: • Given • AD bisects BAC, so D is___ from the sides of BAC • BD bisects ABC, so D is equidistant from the sides of ABC. • Trans. Prop of Equality • Converse of the Angle Bisector Thm. Statements: • ∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCA • DE = DG • DE = DF • DF = DG • D is on the bisector of C. • ________

  26. Given►∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCAProve►The angle bisectors intersect at a point that is equidistant from AB, BC,and CA Reasons: • Given • AD bisects BAC, so D is___ from the sides of BAC • BD bisects ABC, so D is equidistant from the sides of ABC. • Trans. Prop of Equality • Converse of the Angle Bisector Thm. • Givens and Steps 2-4 Statements: • ∆ABC, the bisectors of A, B, and C, DEAB, DFBC, DGCA • DE = DG • DE = DF • DF = DG • D is on the ______ of C. • D is equidistant from Sides of ∆ABC

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