1. 5.1 and 5.2 Midsegments and Bisectors of Triangles

2. Lets look at some formulas first.
Slope:
y2-y1
x2-x1
Midpoint:
Distance:

3. Find the midpoint between two points.
(0.45,7) and (-0.3,-9) ( )
-0.3
+ 0.45 -9
+ 7 , 2 2
(0.075,-1)

4. Find the midpoint between two points.
(6,6) and (19,6) ( ) 19
+ 6 6
+ 6 , 2 2
(12.5, 6)

5. Find the distince between two points.
(6,6) and (19,6) 13

6. Using Midsegments of a Triangle
A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle

7. Theorem 5-1 Triangle Midsegment Theorem
Using Midsegments of a Triangle
Theorem 5-1 Triangle Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle is parallel to the 3rd side and is half as long.

8. Triangle Midsegment Theorem A segment whose endpoints are the
midpoints of two sides of the triangle
Midsegment
Base
Midsegment = ½ of Base
M = ½ b

9. Find the length of the midsegment.
M = ½ b 23
M = ½ • 46 46
M = 23

10. Find the length of the base.
M = ½ b 46 2•
46= ½ b 2• 92
92 = b

11. Find the length of the midsegment and the base.
5•3-1
= 14
M = ½ b
5x - 1
5x - 1 = ½(6x + 10)
5x - 1 = 3x + 5
6x + 10
-3x x
2x - 1 = 5
6•3+10
= 28
+1 +1
2x = 6
x = 3

12. 5.1 Midsegments of Triangles
In ΔEFG, H, J, and K are midpoints. Find HJ, JK, and FG. F
HJ:
JK:
FG: 60 H J 40 G E K
100

13. 5.1 Midsegments of Triangles
AB = 10 and CD = 18. Find EB, BC, and AC A B E D C
EB:
BC:
AC:

14. Using Midsegments of a Triangle
Find JK and AB

15. a) What are the coordinates of Q and R?
Using Midsegments of a Triangle
a) What are the coordinates of Q and R?
b) Why is
c) What is MP? What is QR?

16. a) In XYZ, which segment is parallel to
Using Midsegments of a Triangle
a) In XYZ, which segment is parallel to
b) Is Why?
c) Find YZ and XY

17. Using the Triangle Midsegment Theorem
In the diagram, DE is a midsegment of  ABC. Find the values of x and y.

18. Your Turn: Using Triangle Midsegment Theorem
In the diagram, PQ is a midsegment of  LMN. Find the values of x and y.

19. Using Midsegments of a Triangle
Given: DE = x + 2; BC =
Find DE

20. Using Properties of Midsegments
are midsegments in XYZ. Find the perimeter of XYZ.

21. Using Properties of Midsegments
Given: X, Y, and Z are the midpoints of AB, BC, and AC respectively. AX = 2; XY = 3; BC = 9
Find the perimeter of  ABC.

22. Theorem 5-1: 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
Example 1: Finding Lengths
In XYZ, M, N and P are the midpoints. The Perimeter of MNP is 60. Find NP and YZ.
Because the perimeter is 60, you can find NP.
NP + MN + MP = 60 (Definition of Perimeter)
NP = NP + = 60
NP = 24 22 x P Y M N Z

23. 5.1 Midsegments
Find m<VUZ. Justify your answer.
65° X Y V Z U

24. Example 1 5 ft 16 ft TU = ________ PR = ________
In the diagram, ST and TU are midsegments of triangle PQR. Find PR and TU.
5 ft
16 ft
TU = ________
PR = ________

25. Example 2 14 cm 53 cm ZY = ________ MN = ________
In the diagram, XZ and ZY are midsegments of triangle LMN. Find MN and ZY.
14 cm
53 cm
ZY = ________
MN = ________

26. 5.1 Midsegments
Dean plans to swim the length of the lake, as shown in the photo. He counts the distances shown by counting 3ft strides. How far would Dean swim?
35 strides
118 strides
128 strides x

27. 5.2 Bisectors in Triangles

28. a 90° angles and bisect the side.
Perpendicular Bisectors
A Perpendicular bisector of a side does not have to start at a vertex. It will form
a 90° angles and bisect the side.
Circumcenter

29. When three or more lines intersect at one point, the lines are said to be concurrent.

30. The circumcenter of ΔABC is the center of its circumscribed circle
The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.

31. The incenter is the center of the triangle’s inscribed circle
The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.

32. The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

33. Example 1: Using Properties of Perpendicular Bisectors
DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.
G is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of
∆ABC.
GC = CB
Circumcenter Thm.
GC = 13.4
Substitute 13.4 for GB.

34. MZ is a perpendicular bisector of ∆GHJ.
Check It Out! Example 1a
Use the diagram. Find GM.
MZ is a perpendicular bisector of ∆GHJ.
GM = MJ
Circumcenter Thm.
GM = 14.5
Substitute 14.5 for MJ.

35. KZ is a perpendicular bisector of ∆GHJ.
Check It Out! Example 1b
Use the diagram. Find GK.
KZ is a perpendicular bisector of ∆GHJ.
GK = KH
Circumcenter Thm.
GK = 18.6
Substitute 18.6 for KH.

36. Check It Out! Example 1c
Use the diagram. Find JZ.
Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of
∆GHJ.
JZ = GZ
Circumcenter Thm.
JZ = 19.9
Substitute 19.9 for GZ.

37. Example 2: Finding the Circumcenter of a Triangle
Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).
Step 1 Graph the triangle.

38. Step 2 Find equations for two perpendicular bisectors.
Example 2 Continued
Step 2 Find equations for two perpendicular bisectors.
Since two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of HJ is x = 5, and the perpendicular bisector of HK is y = 3.

39. Step 3 Find the intersection of the two equations.
Example 2 Continued
Step 3 Find the intersection of the two equations.
The lines x = 5 and y = 3 intersect at (5, 3), the circumcenter of ∆HJK.

40. Step 1 Graph the triangle.
Check It Out! Example 2
Find the circumcenter of ∆GOH with vertices G(0, –9), O(0, 0), and H(8, 0) .
Step 1 Graph the triangle.

41. Check It Out! Example 2 Continued
Step 2 Find equations for two perpendicular bisectors.
Since two sides of the triangle lie along the axes, use the graph to find the perpendicular bisectors of these two sides. The perpendicular bisector of GO is y = –4.5, and the perpendicular bisector of OH is
x = 4.

42. Check It Out! Example 2 Continued
Step 3 Find the intersection of the two equations.
The lines x = 4 and y = –4.5 intersect at (4, –4.5), the circumcenter of ∆GOH.

43. A triangle has three angles, so it has three angle bisectors
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle .

44. Key Concepts
The distance from a point to a line is the length of the perpendicular segment from the point to the line.
Example: D is 3 in. from line AB and line AC C B D A 3

45. Example
Using the Angle Bisector Theorem. Find x, FB and FD in the diagram at the right.
Show steps to find x, FB and FD:
7x - 35 A E F C D B
2x + 5

46. Quick Check
a. According to the diagram, how far is K from ray EH? From ray ED? H D C E
(X + 20)O
2xO K 10

47. Quick Check b. What can you conclude about ray EK? 2xO D E C (X + 20)O10

48. Quick Check
c. Find the value of x. H D C E
(X + 20)O
2xO K 10

49. Quick Check
d. Find m<DEH. H D C E
(X + 20)O
2xO K 10

50. Solve for y and find the measure of angle HEF.

51. Find a and b.