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  1. Geometry Chapter 5 Properties of Triangles

  2. Warm-Up 1) Find the coordinates of the midpoint of segment BC. 2) Find slope of segment BC. 3) Find slope of a line perpendicular to segment BC. B A C

  3. Bisectors and Triangles • What is a bisector? • A segment, ray, line, or plane that intersects a segment at its midpoint • Perpendicular bisector • A segment, ray, line, or plane that is perpendicular to a segment at its midpoint. • Perpendicular bisector of a triangle • A bisector perpendicular to one side of a triangle

  4. Perpendicular Bisector and Triangles • Perpendicular Bisector Theorem (5.1-5.2) • If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. • If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. • Equidistant—of equal distance

  5. Theorems with Visuals Here’s how those theorems look:

  6. Using Bisectors • Example 1) In the diagram, line MN is the perpendicular bisector of segment ST. • What segment lengths are equal? • Is Q on line MN? T 12 Q M N 12 S

  7. Similar Example • Example 2) Line PQ is the perpendicular bisector of segment CD. • What segment lengths are equal? • Is T on line PQ? T 7 7 Q C D P

  8. Example 3 Does FA = FB? Why or why not? B A F

  9. Practice P. 267-268: 1-7, 8-10, 16-18, 21-26

  10. Warm-Up • Find the midpoint of segment AB in each problem • 1) A(-2,3), B(4,1) • 2) A(0,5), B(3,6) • 3) A(-4,6), B(3, 10) • Find the equation of the line containing the points. • 4) C(-2,3), D(3,1) • 5) E(7,10), F(-5,-8)

  11. Angle Bisector Theorems (5.3)—If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. (5.4)—If a point is equidistant from two sides of an angle, then it is on the perpendicular bisector of the angle.

  12. Angle Bisector Visual B B A A D D C C

  13. Example Find the value of x. C D B 5x F 2x + 24

  14. Practice P. 268: #11-13, 19-20

  15. Concurrency (but no procurrency) • Concurrent Lines • Three or more lines that intersect in the same point. • Point of concurrency • A point where three or more lines intersect

  16. Perpendicular Bisectors and Concurrency • Concurrency of Perpendicular Bisectors of a Triangle (5.5) • The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle • The point where perp bisectors meet is called the circumcenter. • Note how there are three perpendicular bisectors, and how they meet:

  17. Angles Bisectors and Triangles • Angle Bisector of a Triangle • A bisector for an angle in a triangle • Just as with perpendicular bisectors, there are 3 • One for each angle • Incenter • The point of concurrency for the three angle bisectors

  18. Angle Bisector Theorem • Concurrency of Angle Bisectors of a Triangle (5.6) • The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle

  19. Angle Bisector Example List congruent segments. B E D P C A F

  20. Just In Case • Pythagorean Theorem • If AF = 15 and FP = 8, Then find AP. B E D P C A F

  21. Practice P. 275 #3-4, 10-12 P. 276: #14, 17, 18

  22. Practice Worksheet

  23. Warm-Up • 1) Find the slope using points A(2,5) and C(4,1). • 2) With the slope in #1, write an equation for line segment AC. • 3) Determine the slope of a line perpendicular to AC. • 4) Find the midpoint of segment AC. • 5) Find the equation of a line that has the slope of #3 and passes through the point in #4. • -2 • y = -2x + 9 • -(1/2) • (3,3) • y = -(1/2)x + (9/2)

  24. Introduction to Medians • Find the product in the following: • 1) (5/3) 12 • 2) (2/3) * (6/11) • 3) 6(3/4)

  25. Not to be Confused with Medium • Median of a Triangle • A segment that has its endpoints at a vertex and the midpoint of the side opposite that vertex • Again, there are 3 • 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 A B C D

  26. Using the Theorem • Centroid • The point of concurrency for the medians • If P is the centroid of ΔABC, then AP = (2/3)AF, BP = (2/3)BD, and CP = (2/3)CE A E D P B C F

  27. Practice P. 276 #15, 16 P. 282 #1, 8-12, 17-20

  28. Altitudes • Altitude of a Triangle • The perpendicular segment from a vertex to its opposite side • Every triangle has 3 A B C

  29. Altitude Theorem • Concurrency of Altitudes of a Triangle (5.8) • The lines containing the altitudes of a triangle are concurrent. • Orthocenter • The point of concurrency for the altitudes

  30. Finding the Orthocenter Draw an acute, a right, and an obtuse triangle.

  31. Practice Worksheet

  32. Ticket Out the Door Answer 1-7 on page 282

  33. Warm-Up • Find midpoint between… • 1) …(2,o) and (0,2) 2) …(5,8) and (-3,-4) • 3)…(3,11) and (3,4) 4) …(-2,-6) and (4,-9) • 5)…(4,3) and (8,3)

  34. Warm-Up • 1) Find the slope of the line between, the midpoint between, and the distance between these two points: • A(9,6) and B(8,12) • 2) What is the length of RU? R 6x + 5 8x S U

  35. Midsegments • Midsegment • A segment that connects the midpoints of two sides of a triangle • There are three per triangle • One for each pair of midpoints B D E A C F

  36. Midsegment Theorem (5.9) The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. G L K F H

  37. Example

  38. Example

  39. Example

  40. Example

  41. Example A triangle has coordinates A(3,5), B(7,3), and C(1,1). What are the lengths of its midsegments?

  42. Practice P. 290-291 #12-18, 21-22, 28-29

  43. Warm-Up Solve the inequality. 1) x + 3 < 14 2) 12 > 10 – x 3) 2x + 3 < 4x – 9 Find the measure of the third angle of the triangle. 4) 28°, 59° 5) x°, 2x°

  44. Intro to Triangle Inequalities What do we know about isosceles triangles? What about equilateral triangles?

  45. Triangle Inequalities: One Triangle • Theorem 5.10 • 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 A B C

  46. Triangle Inequalities: One Triangle • Theorem 5.11 • 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 side A B C

  47. Important Theorem • Triangle Inequality (5.13) • The sum of the lengths of any two sides of a triangle is greater than the length of the third side

  48. Another Theorem • Remember the exterior angle of a triangle? How can we find its measure? • Exterior Angle Inequality (5.12) • The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles A C B

  49. It’s the Final Count—I Mean, Theorem (for the chapter) • Hinge Theorem (5.14) • If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second

  50. Warm-Up Match each image with what it is depicting. 1) a) altitude 2) b) angle bisector 3) c) median 4) d) perp. bisector