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Geometry Bisector of Triangles

Geometry Bisector of Triangles. Warm up. Determine whether each point is on the perpendicular bisector of the segment with endpoints S(0,8) and T(4,0). 1) X(0,3) 2) Y(-4,1) 3) Z(-8,-2). yes yes no. Bisector of Triangles.

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Geometry Bisector of Triangles

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  1. Geometry Bisector of Triangles CONFIDENTIAL

  2. Warm up Determine whether each point is on the perpendicular bisector of the segment with endpoints S(0,8) and T(4,0). 1) X(0,3) 2) Y(-4,1) 3) Z(-8,-2) • yes • yes • no CONFIDENTIAL

  3. Bisector of Triangles Since a triangle has three sides, it has three perpendicular bisector. When you construct the perpendicular bisectors, you find that they have an interesting property. CONFIDENTIAL

  4. 1 2 3 B A B P C A C A C Draw a large scalene acute triangle ABC on a piece of patty paper. Fold the perpendicular bisector of each side. Label the point where the three perpendicular bisectors intersect as P. CONFIDENTIAL

  5. When three or more lines intersect at one point, the lines are said to be concurrent. The point ofconcurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle. CONFIDENTIAL

  6. Theorem 2.1 Circumcenter Theorem B The circumcenter of a triangle is equidistant from the vertices of the triangle. PA = PB = PC P C A CONFIDENTIAL

  7. The circumcenter can be inside the triangle, Out the triangle, or on the triangle. P P P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

  8. The circumcenter of ∆ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribedabout thepolygon. B P A C CONFIDENTIAL

  9. Circumcenter Theorem A l n P B C m CONFIDENTIAL

  10. Using Properties of Perpendicular Bisector H 18.6 Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of ∆GHJ. HZ = GZ Circumcenter Thm. HZ = 19.9 Substitute 19.9 for GZ. Z L K 9.5 19.9 G J M 14.5 CONFIDENTIAL

  11. Now you try! • Use the diagram. Find each length. • a) GM b) GK c) JZ H 18.6 Z L K 1a) 14.5 1b) 18.6 1c) 19.9 9.5 19.9 G J M 14.5 CONFIDENTIAL

  12. Finding the Circumcenter of a Triangle. Finding the circumcenter of ∆RSO with vertices R(-6,0), S(0,4), and O (0,0). Step 1 Graph the triangle. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

  13. Step 2 Find equation for two perpendicular bisectors. Since Two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of RO is x = -3, and the perpendicular bisector of OS is y =2. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

  14. Step 3 Find the intersection of the two equations. The lines x = -3 and y = z intersect at (-3,2), the circumcenter of ∆RSO. y X = -4 S Y = 2 (-3,2) x R 4 O CONFIDENTIAL

  15. Now you try! 2) Find the circumcenter of ∆GOH with vertices G(0,-9), O(0,0), and H(8,0). 2) 29 CONFIDENTIAL

  16. Theorem 2.2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. PX = PY = PZ B Z Y P C A X CONFIDENTIAL

  17. Unlike the circumcenter, the incenter is always inside the triangle. P P P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

  18. The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point. B P A C CONFIDENTIAL

  19. Using Properties of Angle Bisectors K A) The distance from V to KL 7.3 V is the incenter of ∆JKL. By the Incenter Theorem, V is equidistant from the sides of ∆JKL. The distance from V to JK is 7.3. So the distance from V to KL is also 7.3 W V 106˚ J L 19˚ Next page: CONFIDENTIAL

  20. B) m/ Vkl JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K m/ kJL = 2m / VJL m/ kJL = 2(19˚) = 38˚ m/ kJL + m/ JLK + m/ JKL = 180˚ 38 + 106 + m/ JKL = 180˚ m/ JKL = 36˚ m/ Vkl = ½ m/ JKL m/ Vkl = ½ (36˚) = 18˚ Substitute the given values. Subtract 144˚ from both sides. 7.3 W KV is the bisector of m/ JKL V Substitute 36˚ for m/ JKL. 106˚ L 19˚ CONFIDENTIAL

  21. Now you try! • 3) QX and RX are angle bisectors ∆PQR. Find each measure. • The distance from X to PQ • m/ PQX Q X 3a) 14.5 3b) 18.6 R P 12˚ Y 19.2 CONFIDENTIAL

  22. Community Application The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing location A, B, and C on the map. Justify your sketch. A Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. C B Next page: CONFIDENTIAL

  23. Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. A C F B CONFIDENTIAL

  24. Now you try! 4) A City plans to build a firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch. Centerville Anverue King Boulevard Third Street CONFIDENTIAL

  25. Now some practice problems for you! CONFIDENTIAL

  26. Assessment SN, TN, and VN are the perpendicular bisectors of ∆PQR. Find each length. 1) NR 2) RV 3) TR 4) QN Q 3.95 S N T 5.64 4.03 P R 5.47 V CONFIDENTIAL

  27. Find the circumcenter of a triangle with the given vertices. • 5) O(0,0), K(0, 12), L(4,0) • 6) A(-7,0), O(0,0),B(0,-10) CONFIDENTIAL

  28. CF and EF are angle bisectors of ∆CDE. Find each measure. 7) The distance from F to CD 8) m/ FED 17˚ C D 54˚ F 42.1 G E CONFIDENTIAL

  29. 9) The designer of the Newtown High School pennant wants the circle around the bear emblem to be as large as possible. Draw a sketch to show where the center of the circle should be located. Justify your sketch. BEARS CONFIDENTIAL

  30. Let’s review Bisector of Triangles Since a triangle has three sides, it has three perpendicular bisector. When you construct the perpendicular bisectors, you find that they have an interesting property. CONFIDENTIAL

  31. 1 2 3 B A B P C A C A C Draw a large scalene acute triangle ABC on a piece of patty paper. Fold the perpendicular bisector of each side. Label the point where the three perpendicular bisectors intersect as P. CONFIDENTIAL

  32. When three or more lines intersect at one point, the lines are said to be concurrent. The point ofconcurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle. CONFIDENTIAL

  33. Theorem 2.1 Circumcenter Theorem B The circumcenter of a triangle is equidistant from the vertices of the triangle. PA = PB = PC P C A CONFIDENTIAL

  34. The circumcenter can be inside the triangle, Out the triangle, or on the triangle. P P P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

  35. The circumcenter of ∆ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribedabout thepolygon. B P A C CONFIDENTIAL

  36. Circumcenter Theorem A l n P B C m CONFIDENTIAL

  37. Using Properties of Perpendicular Bisector H 18.6 Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of ∆GHJ. HZ = GZ Circumcenter Thm. HZ = 19.9 Substitute 19.9 for GZ. Z L K 9.5 19.9 G J M 14.5 CONFIDENTIAL

  38. Finding the Circumcenter of a Triangle. Finding the circumcenter of ∆RSO with vertices R(-6,0), S(0,4), and O (0,0). Step 1 Graph the triangle. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

  39. Step 2 Find equation for two perpendicular bisectors. Since Two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of RO is x = -3, and the perpendicular bisector of OS is y =2. y X = -4 S Y = 2 (-3,2) x R 4 O Next page: CONFIDENTIAL

  40. Step 3 Find the intersection of the two equations. The lines x = -3 and y = z intersect at (-3,2), the circumcenter of ∆RSO. y X = -4 S Y = 2 (-3,2) x R 4 O CONFIDENTIAL

  41. Theorem 2.2 Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. PX = PY = PZ B Z Y P C A X CONFIDENTIAL

  42. Unlike the circumcenter, the incenter is always inside the triangle. P P P Acute triangle Obtuse triangle Right triangle CONFIDENTIAL

  43. The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point. B P A C CONFIDENTIAL

  44. Using Properties of Angle Bisectors K A) The distance from V to KL 7.3 V is the incenter of ∆JKL. By the Incenter Theorem, V is equidistant from the sides of ∆JKL. The distance from V to JK is 7.3. So the distance from V to KL is also 7.3 W V 106˚ J L 19˚ Next page: CONFIDENTIAL

  45. B) m/ Vkl JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K m/ kJL = 2m / VJL m/ kJL = 2(19˚) = 38˚ m/ kJL + m/ JLK + m/ JKL = 180˚ 38 + 106 + m/ JKL = 180˚ m/ JKL = 36˚ m/ Vkl = ½ m/ JKL m/ Vkl = ½ (36˚) = 18˚ Substitute the given values. Subtract 144˚ from both sides. 7.3 W KV is the bisector of m/ JKL V Substitute 36˚ for m/ JKL. 106˚ L 19˚ CONFIDENTIAL

  46. Community Application The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing location A, B, and C on the map. Justify your sketch. A Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. C B Next page: CONFIDENTIAL

  47. Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. A C F B CONFIDENTIAL

  48. You did a great job today! CONFIDENTIAL

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