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Chapter 5

Chapter 5 . Relationships Within Triangles. Warm Up:. Draw and cut out a triangle, any kind. Fold one of the vertices such that it touches the opposite side and fold. Measure the lengths a - f. Midsegments. The segment connecting the midpoints of two sides.

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Chapter 5

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  1. Chapter 5 Relationships Within Triangles

  2. Warm Up: • Draw and cut out a triangle, any kind. • Fold one of the vertices such that it touches the opposite side and fold. • Measure the lengths a - f.

  3. Midsegments • The segment connecting the midpoints of two sides Triangles have three midsegments

  4. Triangle Midsegment Theorem • If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half its length.

  5. Example: Q and P are midpoints of two sides of the triangle. Find x and the perimeter of the triangle.

  6. In ∆XYZ, M, N, and P are midpoints. The perimeter of ∆MNP is 60. Find NP and the perimeter of the triangle.

  7. 5.2 Bisectors in Triangles

  8. Triangles play a key role in relationships involving perpendicular bisectors and angle bisectors. B A

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

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

  11. C 5 A B 6 D Example One

  12. Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

  13. Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

  14. Examples: Solve for the given variables.

  15. Example 2: Find FD C D B 5x 2x + 24 F FD = 2x + 24 = 2(8) +24 = 16 + 24 = 40 5x = 2x + 24 3x = 24 x = 8

  16. 1. Find the value of x. 2. Find CG. 3. Find the perimeter of quadrilateral ABCG. 6 8 48

  17. Examples: Find the perimeter.

  18. 5.3 Concurrent Lines

  19. Concurrent When three or more lines intersect in one point Point of Concurrency

  20. Median The endpoints are a vertex of the triangle and the midpoint of the opposite side

  21. Theorem 5-8 The medians of a triangle are concurrent at a point (the centroid) that is two thirds the distance from each vertex to the midpoint of the opposite side.

  22. Find AB and AC if C is the centroid. A C 12 B

  23. Checkpoint Use the Centroid of a Triangle The centroid of the triangle is shown. Find the lengths. 4. Find BE and ED, given BD=24. Find JG and KG, given JK=4. 5. PQ=10; PN=30 BE=16; ED=8 JG=12; KG=8 ANSWER ANSWER ANSWER Find PQ and PN, given QN=20. 6.

  24. Altitude The perpendicular segment from a vertex to the line containing the opposite side (aka the height)

  25. Altitude of a Triangle • Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle, or it may lie outside the triangle. • The point of concurrency for the altitudes is called the orthocenter.

  26. * Theorem 5-6 • The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. • The point of concurrency for the perpendicular bisectors is called the circumcenter.

  27. Circumscribed Circles

  28. * Theorem 5-7 • The bisectors of the angles of a triangle are concurrent at a point equidistant from its sides. • The point of concurrency for the angle bisectors is called the incenter.

  29. Inscribed Circles

  30. 5-5 Triangle Inequalities

  31. Find the m<1. Complete the inequality statement: > >

  32. Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

  33. Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

  34. Theorem 5-11: If two sides of a triangle are not congruent, then the longer side lies opposite the larger angle. Note: This is similar to 5-10, but reverses the order of the longer side and larger angle.

  35. Checkpoint Order Angle Measures and Side Lengths Name the angles from largest to smallest. 1. ANSWER N;L;M 2. ANSWER Q;R;P 3. ANSWER U;S;T

  36. Checkpoint Name the sides from longest to shortest. Order Angle Measures and Side Lengths 4. ANSWER GH;JG;JH 5. ANSWER DE;EF;DF AC;AB;BC 6. ANSWER

  37. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

  38. Checkpoint Use the Triangle Inequality Can the side lengths form a triangle? Explain. 5, 7, 13 7. ANSWER No; 5 + 7 < 13. 6, 9, 12 8. Yes; 6 + 9 > 12, 6 + 12 > 9,and 9 + 12 > 6. ANSWER 10, 15, 25 9. ANSWER No; 10 + 15 = 25.

  39. The lengths of two sides of a triangle are given, what are the possible lengths for the third side?6”, 10” The difference of the two given sides is the minimum value that x can be. The sum of the two given sides is the maximum value. Hint: what are the three cases? The limits: 4 < x < 16

  40. Coordinate Proofs: Given: R is the midpoint of S is the midpoint of Prove: Place the triangle in a convenient spot on the coordinate plane (typically with a vertex at (0, 0).

  41. 2. Choose variables for the coordinates of the vertices.

  42. To find the coordinates of R and S, use the midpoint formula: Q (b, 0) and P (c, a) : O (0, 0) and P (c, a) :

  43. To show , prove their slopes are equal. 3. Use the coordinates to prove the desired properties. = 0 = 0 Therefore, since the slopes are the same, the lines are parallel.

  44. To show , prove using their distances. 3. Use the coordinates to prove the desired properties.

  45. Therefore, since OQ = b and RS = , you have proven

  46. Warm Up: Write a coordinate proof showing opposite sides of a rectangle are parallel.

  47. Given: Prove:

  48. Objective One • The first objective in this section is to use inequalities involving angles of triangles.

  49. Comparison Property of Equality

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