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5-1. Special Segments in Triangles. I. Triangles have four types of special segments:. A. Perpendicular bisector. Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment every triangle has 3 perpendicular bisectors.

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  1. 5-1 Special Segments in Triangles

  2. I. Triangles have four types of special segments:

  3. A. Perpendicular bisector • Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment • every triangle has 3 perpendicular bisectors

  4. Examples of perpendicular bisectors In an acute triangle, the perpendicular bisectors meet inside the triangle. In a right triangle, the perpendicular bisectors meet on the triangle. In an obtuse triangle, the perpendicular bisectors meet outside the triangle.

  5. Circumcenter • The point where all the perp. Bisectors meet in a triangle

  6. B. Median A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. D

  7. PROPERTIES OF MEDIANS • every triangle has 3 medians • the medians always meet inside the triangle

  8. Centroid • Point where all the medians meet

  9. C. Altitude

  10. The three altitudes intersect at G,a point inside the triangle.

  11. In right triangle ABC, segment AB and segment BC are two of the altitudes of the triangle A B C

  12. Orthocenter • Point where all the altitudes meet

  13. D. Angle bisector

  14. -there are 3 angle bisectors in every triangle. -any point on an angle bisector of an angle is equidistant (same distance) from the sides of the angle. -any point on or in the interior of an angle and equidistant from the sides of the angle lies on the angle bisector. -if a triangle is isosceles, the bisector of the vertex angle is also a median and an altitude. -The angle bisectors always meet inside the triangle.

  15. Incenter • Point where all the angle bisectors meet

  16. 1. Given: < F = 80, < E = 30DG bisects < EDFProve: < DGE = 115 F G E D

  17. Find x and < 2.MS is an altitude of  MNQ< 1 = 3x + 11< 2 = 7x + 9 M R 1 2 Q N S

  18. MS is a median of  MNQ.QS = 3a -14 SN = 2 a + 1<MSQ = 7a + 1 Find a.Is MS an altitude? M R 1 2 Q N S

  19. 4. Always, sometimes, never • 3 medians intersect in the interior • 3 altitudes intersect at a vertex • 3 angle bisectors intersect on the exterior • 3 perpendicular bisectors intersect on the exterior

  20. 5. Triangle ABC has vertices A(-3, 10) B(9, 2) and C(9, 15) • Determine the coordinates of P on segment AB so that segment CP is • a median of the triangle • Determine if segment CP is also an altitude of the triangle • P is the midpoint of segment AB • (-3 + 9)/2 (10 + 2)/2 • The midpoint is P(3, 6) • The slope of segment CP is (6 – 15)/(3 – 9) = 3/2 • The slope of segment AB is (2 – 10)/(9 - - 3) = -2/3 • Since the slopes are opposite reciprocals, • CP is perpendicular to AB. • Therefore, segment CP is an altitude C B A B

  21. A 6. Find the indicated information: a. b. P D B C Find BD if AC is an altitude and m<ACD = 3x + 30, BC = x + 4 and CD = 2x + 8 Q R S W Find PQ if segment PS is a median of the triangle QS = x + 5 SR = 3x – 17 And PQ = 2x -8 c. X Y Z Find m<XWZ if m<XWY = 2x – 4 and m<XWZ = 5x – 12. Segment WY is an angle bisector.

  22. Solutions to example 6: • Since segment PS is a median, QS = SR. So, x + 5 = 3x – 17. • Then 5 = 2x – 17 22 = 2x 11 = x • Finally PQ = 2x – 8 2(11) – 8 PQ = 14 b. If BC is an altitude, then <ACB is a right angle. Then, 3x + 30 = 90. 3x = 60 x = 20 Finally BD = BC + CD x + 4 + 2x + 8 3x + 12 3(20) +12 BD = 72 c. Since segment WY is an angle bisector, it divides the angle in half. So m<XWZ = 2m<XWY. Therefore 5x – 12 = 2(2x - 4) 5x – 12 = 4x - 8 x – 12 = -8 x = 4 So m<XWZ = 5x – 12 5(4) – 12 m<XWZ = 8

  23. Example 7: Write at least one conclusion that can be made from each of the following statements. S • Segment SM is an altitude to segment RE • SN = NE • M is equidistant from R and E, and <RMS • is a right angle • m<ERN = m<SRN • segment EL is perpendicular to segment SR N L E R M • Sample answers: • Segment SM is perpendicular to segment RE • <SMR and <SME are right angles • b. segment RN is a median and N is the midpoint • c. Segment SM is a perpendicular bisector • segment SM is a median • segment SM is an altitude • Segment RN is an angle bisector • Segment EL is an altitude and <ELR and <ELS are right angles

  24. Your turn!

  25. 1. The segment that bisects an angle of the  and has one endpoint at a vertex of the  and the other endpoint at another point on the  is called the _________.

  26. 2.  A _________ is a line or segment that passes through the midpoint of a side of a  and is perpendicular to that side.

  27. 3. Triangle XYZ  has vertices X(-1, 1), Y(3, 9), and Z(6, -2). Determine thecoordinates of point W on so that is a median of the triangle.

  28. 4. Triangle CPR has vertices C(15, 1), P(9, 11), and R(2, 1). Determine the coordinates of point A on so that is a median of triangle CPR.

  29. 5-2 Inequalities for the sides and angles of a triangle • I. Theorem • The angle opposite the larger side is always bigger than the angle opposite the shorter side of any triangle

  30. II. Theorem • If one angle is larger than another angle, then the side opposite it is also larger than the side opposite the smaller angle.

  31. III. Theorem • The perpendicular segment from a point to a line is the shortest segment from the point to the line. • Corollary The perp. Segment from a point to a plane is the shortest segment also.

  32. IV.Examples • 1. Refer to the figure in example 1 of book. • Given: angle A is greater than angle D • Prove: BD is greater than AB

  33. 2. Draw triangle JKL with J(-4,2) K (4,3) L (1,-3). List the angles from greatest to least measure.

  34. 5-3 Indirect Proof

  35. Indirect Proof Steps • Assume that the conclusion is false. • Show that this leads to a contradiction of a known property, rule, etc… • Point this out and since it is false, it follows that the conclusion must be true. You are proving that the contrapositive of the conditional is true, therefore the original conditional must be true

  36. II. A. Exterior Angle Inequality Theorem • If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. Recall that the sum of the two remote interior angles is equal to the exterior. Since neither of the interior angles are zero, the exterior angle will always be bigger than either of them.

  37. B. Definition of an inequality • For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c.

  38. Properties of Inequalities for Real Numbers p. 254 in book Comparison Property: one of three things has to be true a<b a>b a=b Transitive Property: If a<b and b<c, then a<c If a>b and b>c, then a>c

  39. Addition and Subtraction Property: if a>b then a+c>b+c and a-c>b-c if a<b then a+c<b+c and a-c<b-c Multiplication and Division Property if c>o (positive)and a>b then ac>bc and a/c>b/c if c>o (positive)and a<b then ac<bc and a/c<b/c if c<o (negative) and a>b then ac<bc and a/c<b/c if c<o (negative) and a<b then ac>bc and a/c>b/c REMEMBER:if you multiply or divide an inequality by a negative number it switches the direction of the inequality

  40. III. Examples • 1.  Which assumption would you make to start an indirect proof of the statement “two acute angles are congruent”.

  41. 2.   Which assumption would you make to start an indirect proof of the following statements? • Bob took the dog for a walk • EF is not a perpendicular bisector • 3x = 4 y + 1 • < 1 is less than or equal to < 2

  42. 3. Name the property that justifies if a < b, then a + c < b + c. • 4. Name the property that justifies that if a is less than b, then a cannot be greater than b. • 5. Given: 2 y + 8 = 16 Prove: y ≠ 5

  43. J • 6. Given: JKL with side lengths 3, 4, 5 Prove: < K < < L K L

  44. Refer to page 255 in your textbook and try 5 –13(hint for 8 – 10: use the exterior angle inequality theorem-the exterior angle is greater than the remote interior angles)

  45. Answers to p 255-2565 -13 5. Assume lines l and m do not intersect at x 6. Assume: If the alt int <‘s formed by two parallel lines and a transversal are congruent, then the lines are not parallel. 7. Assume Sabrina did not eat the leftover pizza 8. <3, <7, <5, <6 9. <4 and <8 10. <8 > <7 because <7 is part of a remote interior angle for the exterior <8 11. Division 12. Addition 13. Transitive

  46. 5-4 The Triangle Inequality • I. Theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

  47. II. Examples • 1. If Mrs. Ewing gave Elizabeth four pieces of tubing measuring 6 m, 7 m, 9 m, and 16 m, how many different triangles could she make?

  48. 2. What are the possible lengths for the third side of a triangle with two sides of 8 and 13?

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