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Chapter 5

Chapter 5. Properties of Triangles. Chapter 5 Objectives. Identify a perpendicular bisector Identify characteristics of angle bisectors Visualize concurrency points of a triangle Compare measurements of a triangle Display the midsegment of a triangle Utilize the triangle inequality theorem

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Chapter 5

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  1. Chapter 5 Properties of Triangles

  2. Chapter 5 Objectives • Identify a perpendicular bisector • Identify characteristics of angle bisectors • Visualize concurrency points of a triangle • Compare measurements of a triangle • Display the midsegment of a triangle • Utilize the triangle inequality theorem • Create an indirect proof

  3. Lesson 5.1 Perpendiculars and Bisectors

  4. Lesson 5.1 Objectives • Define perpendicular bisector • Utilize the Perpendicular Bisector Theorem and its converse

  5. Perpendicular Bisector • A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector.

  6. Equidistant • In order for an object to be equidistant from two or more locations, the following must be true: • The distance to each object must be equal. • The segment drawn from the object must intersect each location at the same angle.

  7. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Theorem 5.1:Perpendicular Bisector Theorem

  8. Theorem 5.2:Perpendicular Bisector Converse • If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

  9. Example 1 Tell whether there is enough information that C lies on the perpendicular bisector of segment AB.Explain. Yup! Yup! C is equidistant from A and B C is equidistant from A and B

  10. If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. Theorem 5.3:Angle Bisector Theorem

  11. If a point is on the interior of an angle and it is equidistant from the two sides of the angle, then it lies on the bisector of an angle. Theorem 5.4:Angle Bisector Converse

  12. Example 2 Can you conclude that ray BD bisects ABC?Explain. Yup! Nope! We do not know the angles at which the segments intersect the sides of ABC. D is equidistant from A and C

  13. Lesson 5.1 Homework • In Class • 1-7 • p268-271 • HW • 8-13, 16-26, 41-52 • Due Tomorrow

  14. Lesson 5.2 Bisectors of a Triangle

  15. Lesson 5.2 Objectives • Define concurrency • Identify the concurrent points inside triangles. • Identify perpendicular and angle bisectors in a triangle. • Differentiate between circumcenter and incenter

  16. Perpendicular Bisectors of a Triangle • A perpendicular bisector of a triangle is any segment or ray or line that is perpendicular to the midpoint of any side of a triangle.

  17. Concurrent lines exist when 3or more lines, segments, or rays intersect at a common point. The point at which the concurrent lines intersect is called the point of concurrency. Concurrency

  18. Theorem 5.5:Perpendicular Bisectors of a Triangle • The perpendicular bisectors of a triangle will intersect to form a point of concurrencyequidistant from the vertices. Hint:If the segment is perpendicularto a side, then it isequidistant to the vertices.

  19. Circumcenter • The point of concurrency of perpendicular bisectors in a triangle is called the circumcenter of a triangle. • It is called this because it forms the center of a circle that is drawn connecting the vertices of the triangle. • Notice the vertices of the triangle lie on the circumference of the circle. • Thus the name circum-center.

  20. Example 3 Find the following quantities: • MO • 26.8 • PR • 26 • MN • 40 • SP • 22 • MP • 44

  21. Inside Out Note: All lines drawn must be perpendicular bisectors of the triangle sides.

  22. Theorem 5.6:Angle Bisectors of a Triangle • The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. Hint:When angles are equal, then the distance to the side is equal. Hint:But the perpendicular segments are not bisectors.

  23. Incenter • The point of concurrency of the angle bisectors is called the incenter of the triangle. • It is called this because it creates the center of a circle formed by touching each side of the triangle once. • Notice the circle formed is inside the triangle. • Thus the name in-center.

  24. Example 4 Point T is the incenter of PQR. Find ST. 15

  25. Example 5 • Three snack carts sell frozen yogurt at locations A, B, and C. The distributor for the snack carts wants to build a warehouse that is equal distance to all three carts. • Describe how the distributor could find a location for the warehouse. • Show where the warehouse should be built on the map. Mr. Lent’s Ice Cream Warehouse Use the perpendicular bisectors of a triangle to determine the circumcenter of the three locations. The circumcenter is equidistant from all vertices of a triangle.

  26. Lesson 5.2 Homework • In Class • 1-4 • p275-278 • HW • 5-21, 24-28 • Due Tomorrow

  27. Lesson 5.3 Medians and Altitudes of Triangles

  28. Lesson 5.3 Objectives • Define a median of a triangle • Identify a centroid of a triangle • Define the altitude of a triangle • Identify the orthocenter of a triangle

  29. A B C Triangle Medians • A median of a triangle is a segment that does the following: • Contains one endpoint at a vertex of the triangle, and • Contains the other endpoint at the midpoint of the opposite side of the triangle. D

  30. Remember: All mediansintersect the midpointof the opposite side. Centroid • When all three medians are drawn in, they intersect to form the centroid of a triangle. • This special point of concurrency is the balance point for any evenly distributed triangle. • In Physics, we would call it thecenter of mass. Obtuse Acute Right

  31. Theorem 5.7:Concurrency of Medians of a Triangle • The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. • The centroid is 2/3 the distance from any vertex to the opposite side. AP = 2/3AE 2/3BF BP = 2/3BF 2/3AE CP = 2/3CD 2/3CD

  32. Example 6 S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the following: • RV • 6 • RU • 6 • 4 is 2/3 of 6 • Divide 4 by 2 and then muliply by 3. Works everytime!! • SU • 2 • RW • 12 • TS • 6 • 6 is 2/3 of 9 • SV • 3

  33. Altitudes • An altitude of a triangle is the perpendicularsegment from a vertex to the opposite side. • It does not bisectthe angle. • It does not bisect the side. • The altitude is often thought of as the height. • While true, there are 3altitudesin every triangle but only 1height!

  34. Orthocenter • The three altitudesof a triangle intersect at a point that we call the orthocenter of the triangle. • The orthocenter can be located: • inside the triangle • outside the triangle, or • on one side of the triangle Obtuse Right Acute The orthocenter of a right triangle will always be located at the vertex that forms the right angle.

  35. Theorem 5.8:Concurrency of Altitudes of a Triangle • The lines containing the altitudes of a triangle are concurrent.

  36. Example 7 Is segment BD a median, altitude, or perpendicular bisector of ABC? Hint: It could be more than one! Perpendicular Bisector Altitude None Median

  37. Lesson 5.3 Homework • In Class • 1-7 • p282-284 • HW • 8-23, 39-45 • Due Tomorrow • Quiz Tuesday • November 20

  38. Lesson 5.4 Midsegment Theorem of Triangles

  39. Lesson 5.4 Objectives • Create the midsegment of a triangle • Identify the characteristics of a midsegment of a triangle

  40. Midsegment of a Triangle • So far we have studied 4 types of special segments of triangles. • Perpendicular Bisector • Angle Bisector • Median • Altitude • It just so happens that all of these intersect only one side at a time. • And three of the four intersect an vertex and a side. • Another type of special segment is one that connects the midpoints of the sides of a triangle. • This special segment is called the midsegment of a triangle. • Notice there are 3midsegments in every triangle.

  41. Theorem 5.9:Midsegment Theorem of a Triangle • The segment connecting the midpoints of the two sides of a triangle is: • Parallel to the third side • Half the length of the third side • The side it is parallel to DE = 1/2AC Segment DE // Segment AC

  42. Example 8 Segment MP is the midsegment of LNO. Find x MP = ½NO MP = ½NO P is the midpoint x = ½(16) 7 = ½(x) x = 4 x = 8 x = 14

  43. Example 9 Fill in the following • Segment GJ is parallel to ________. • segment DF • Segment EJ is congruent to _________. • segment JF • Segment DE is parallel to __________. • segment KJ • If EF = 18, then GK = _____. • 9 • If JK = 13, then ED = _____. • 26

  44. Lesson 5.4 Homework • In Class • 1-11 • p290-293 • HW • 12-18, 21-29, 39-49 odds • Due Tomorrow

  45. Lesson 5.5 Inequalities in One Triangle

  46. Lesson 5.5 Objectives • Compare angle sizes based on side lengths • Utilize the Triangle Inequality Theorem

  47. Theorem 5.10:Side Lengths of a Triangle Theorem • If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. • Basically, the larger the side, the larger the angle opposite that side. 2nd Largest Angle Longest side Smallest Side Smallest Angle Largest Angle 2nd Longest Side

  48. Theorem 5.11:Angle Measures of a Triangle Theorem • If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. • Basically, the larger the angle, the larger the side opposite that angle. 2nd Largest Angle Longest side Smallest Side Smallest Angle Largest Angle 2nd Longest Side

  49. Example 10 Name the smallest and largest angle. Largest Smallest Largest Smallest Largest Smallest

  50. Example 11 Name the smallest and largest side. Largest Smallest Largest Smallest

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