1. 5-1: Special Segments in Triangles
Expectation:
G1.2.5: Solve multi-step problems and construct proofs about the properties of medians, altitudes and perpendicular bisectors to the sides of a triangle and the angle bisectors of a triangle.
4/25/2017
5-1: Special Segments in Triangles

2. 5-1: Special Segments in Triangles
You have a piece of string 120 cm long. What is the area of the largest square you can enclose?
4/25/2017
5-1: Special Segments in Triangles

3. 5-1: Special Segments in Triangles
What is the length of the hypotenuse of the isosceles triangle below? 20 40
800
20√2
40√2 20
4/25/2017
5-1: Special Segments in Triangles

4. 5-1: Special Segments in Triangles
Median of a Triangle
Defn: Median of a Triangle: A segment is a median of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is the midpoint of the side opposite that vertex.
4/25/2017
5-1: Special Segments in Triangles

5. 5-1: Special Segments in Triangles
Medians of a Triangle
Every triangle has 3 medians.
4/25/2017
5-1: Special Segments in Triangles

6. 5-1: Special Segments in Triangles
Medians of a Triangle
Every triangle has 3 medians.
4/25/2017
5-1: Special Segments in Triangles

7. 5-1: Special Segments in Triangles
Medians of a Triangle
Every triangle has 3 medians.
4/25/2017
5-1: Special Segments in Triangles

8. 5-1: Special Segments in Triangles
Centroids
The medians of a triangle will always intersect at the same point - the centroid. The centroid of a triangle is located 2/3 of the distance from the vertex to the midpoint of the opposite side.
4/25/2017
5-1: Special Segments in Triangles

9. 5-1: Special Segments in Triangles
Centroid
centroid
4/25/2017
5-1: Special Segments in Triangles

10. 5-1: Special Segments in Triangles
Centroid
4/25/2017
5-1: Special Segments in Triangles

11. 5-1: Special Segments in Triangles
Points U, V, and W are the midpoints of YZ, XZ and XY respectively. Find a, b, and c.
4/25/2017
5-1: Special Segments in Triangles

12. Perpendicular Bisectors of a Triangle
Defn: Perpendicular Bisector of a Triangle: A segment is a perpendicular bisector of a triangle iff it is the perpendicular bisector of a side of the triangle.
4/25/2017
5-1: Special Segments in Triangles

13. Perpendicular Bisectors of a Triangle
Every triangle has 3 perpendicular bisectors.
4/25/2017
5-1: Special Segments in Triangles

14. Perpendicular Bisectors of a Triangle
4/25/2017
5-1: Special Segments in Triangles

15. Perpendicular Bisectors of a Triangle
4/25/2017
5-1: Special Segments in Triangles

16. Perpendicular Bisectors of a Triangle
4/25/2017
5-1: Special Segments in Triangles

17. 5-1: Special Segments in Triangles
The 3 perpendicular bisectors of any triangle will intersect at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter and is the center of a circle that contains all 3 vertices of the triangle.
4/25/2017
5-1: Special Segments in Triangles

18. Perpendicular Bisector Theorem
A point lies on the perpendicular bisector of a segment iff it is equidistant from the endpoints of the segment.
4/25/2017
5-1: Special Segments in Triangles

19. Perpendicular Bisector Theorem
If AC = BC, then C A B
C is on the perpendicular bisector of AB.
4/25/2017
5-1: Special Segments in Triangles

20. Perpendicular Bisector Theorem
If l is the perpendicular bisector of AB,
then AC = BC and AD = BD.
4/25/2017
5-1: Special Segments in Triangles

21. 5-1: Special Segments in Triangles
Lines s, t, and u are perpendicular bisectors of ∆FGH and meet at J. If JG = 4x + 3, JH = 2y - 3, JF = 7 and HI = 3z - 4, find x, y, and z. t H s u G I J 1 1 F
4/25/2017
5-1: Special Segments in Triangles

22. Altitudes of Triangles
Defn: Altitude of a Triangle: A segment is an altitude of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is on the line containing the opposite side such that the segment is perpendicular to line.
4/25/2017
5-1: Special Segments in Triangles

23. Altitudes of Triangles
Every triangle has 3 altitudes that will always intersect in the same point.
4/25/2017
5-1: Special Segments in Triangles

24. Altitudes of Triangles
If the triangle is acute, then the altitudes are all in the interior of the triangle.
4/25/2017
5-1: Special Segments in Triangles

25. Altitudes of Triangles
If the triangle is a right triangle, then one altitude is in the interior and the other 2 altitudes are the legs of the triangle.
4/25/2017
5-1: Special Segments in Triangles

26. Altitudes of Triangles
If the triangle is an obtuse triangle, then one altitude is in the interior and the other 2 altitudes are in the exterior of the triangle.
4/25/2017
5-1: Special Segments in Triangles

27. Altitudes of Triangles
4/25/2017
5-1: Special Segments in Triangles

28. ZC is an altitude, m∠CYW = 9x + 38 and m∠WZC = 17x. Find m∠WZC.
4/25/2017
5-1: Special Segments in Triangles

29. Angle Bisectors of Triangles
Defn: Angle Bisector of a Triangle: A segment is an angle bisector of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is any other point on the triangle such that the segment bisects an angle of the triangle.
4/25/2017
5-1: Special Segments in Triangles

30. Angle Bisectors of Triangles
Every triangle has 3 angle bisectors which will always intersect in the same point - the incenter. The incenter is the same distance from all 3 sides of the triangle. The incenter of a triangle is also the center of a circle that will intersect each side of the triangle in exactly one point.
4/25/2017
5-1: Special Segments in Triangles

31. Angle Bisectors of Triangles
4/25/2017
5-1: Special Segments in Triangles

32. Angle Bisectors of Triangles
4/25/2017
5-1: Special Segments in Triangles

33. Angle Bisectors of Triangles
4/25/2017
5-1: Special Segments in Triangles

34. 5-1: Special Segments in Triangles
RU is an angle bisector, m∠RTU = 13x – 24, m∠TRS = 12x – 34 and m∠RUS = 92. Determine m∠RSU. Is RU ⊥ TS? R S T U
4/25/2017
5-1: Special Segments in Triangles

35. Angle Bisector Theorem
If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
4/25/2017
5-1: Special Segments in Triangles

36. Angle Bisector Theorem
If D is on the bisector of ∠ABC, then X D B Y C
DX = DY.
4/25/2017
5-1: Special Segments in Triangles

37. Angle Bisector Converse Theorem
If a point is in the interior of an angle and is equidistant from the sides of the angle, then the point lies on the bisector of the angle.
4/25/2017
5-1: Special Segments in Triangles

38. Angle Bisector Converse TheoremX
If WX = WY, then W is on the bisector of ∠XYZ. W Y Z
4/25/2017
5-1: Special Segments in Triangles

39. 5-1: Special Segments in Triangles
In ΔABC below, AB ≅ BC and AD bisects ∠BAC. If the length of BD is 3(x + 2) units and BC = 42 units, what is the value of x? 5 6 12 13 A C D B
4/25/2017
5-1: Special Segments in Triangles

40. 5-1: Special Segments in Triangles
Assignment
Worksheet 5-1
4/25/2017
5-1: Special Segments in Triangles