1. Properties of Triangles
Chapter 5
Properties of Triangles

2. Chapter 5 Objectives Identify a perpendicular bisector
Identify characteristics of angle bisectors
Visualize concurrency points of a triangle
Compare measurements of a triangle
Display the midsegment of a triangle
Utilize the triangle inequality theorem
Create an indirect proof

3. Perpendiculars and Bisectors
Lesson 5.1
Perpendiculars
and
Bisectors

4. Lesson 5.1 Objectives Define perpendicular bisector
Utilize the Perpendicular Bisector Theorem and its converse

5. Perpendicular Bisector
A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector.

6. Equidistant
In order for an object to be equidistant from two or more locations, the following must be true:
The distance to each object must be equal.
The segment drawn from the object must intersect each location at the same angle.

7. Theorem 5.1: Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

8. Theorem 5.2: Perpendicular Bisector Converse
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

9. Example 1
Tell whether there is enough information that C lies on the perpendicular bisector of segment AB. Explain.
Yup!
Yup!
C is equidistant from A and B
C is equidistant from A and B

10. Theorem 5.3: Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

11. Theorem 5.4: Angle Bisector Converse
If a point is on the interior of an angle and it is equidistant from the two sides of the angle, then it lies on the bisector of an angle.

12. Example 2 Can you conclude that ray BD bisects ABC? Explain. Yup!
Nope!
We do not know the angles
at which the segments
intersect the sides of ABC.
D is equidistant from A and C

13. Lesson 5.1 Homework In Class HW Due Tomorrow 1-7 8-13, 16-26, 41-52p HW
8-13, 16-26, 41-52
Due Tomorrow

14. Bisectors of a Triangle
Lesson 5.2
Bisectors
of a
Triangle

15. Lesson 5.2 Objectives Define concurrency
Identify the concurrent points inside triangles.
Identify perpendicular and angle bisectors in a triangle.
Differentiate between circumcenter and incenter

16. Perpendicular Bisectors of a Triangle
A perpendicular bisector of a triangle is any segment or ray or line that is perpendicular to the midpoint of any side of a triangle.

17. Concurrency
Concurrent lines exist when 3 or more lines, segments, or rays intersect at a common point.
The point at which the concurrent lines intersect is called the point of concurrency.

18. Theorem 5.5: Perpendicular Bisectors of a Triangle
The perpendicular bisectors of a triangle will intersect to form a point of concurrency equidistant from the vertices.
Hint: If the segment is perpendicular to a side, then it is equidistant to the vertices.

19. Circumcenter
The point of concurrency of perpendicular bisectors in a triangle is called the circumcenter of a triangle.
It is called this because it forms the center of a circle that is drawn connecting the vertices of the triangle.
Notice the vertices of the triangle lie on the circumference of the circle.
Thus the name circum-center.

20. Example 3 Find the following quantities: MO PR MN SP MP 26.8 26 40 2244

21. Inside Out Type of Triangle Picture Point of Concurrency Inside
Acute Triangle
Right Triangle
Obtuse Triangle
Picture
Point of Concurrency
Inside
On One Side
Outside
Note: All lines drawn must be perpendicular bisectors of the triangle sides.

22. Theorem 5.6: Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.
Hint: When angles are equal, then the distance to the side is equal.
Hint: But the perpendicular segments are not bisectors.

23. Incenter
The point of concurrency of the angle bisectors is called the incenter of the triangle.
It is called this because it creates the center of a circle formed by touching each side of the triangle once.
Notice the circle formed is inside the triangle.
Thus the name in-center.

24. Example 4
Point T is the incenter of PQR.
Find ST. 15

25. Example 5
Three snack carts sell frozen yogurt at locations A, B, and C. The distributor for the snack carts wants to build a warehouse that is equal distance to all three carts.
Describe how the distributor could find a location for the warehouse.
Show where the warehouse should be built on the map.
Mr. Lent’s Ice Cream Warehouse
Use the perpendicular bisectors of a triangle to determine the circumcenter of the three locations. The circumcenter is equidistant from all vertices of a triangle.

26. Lesson 5.2 Homework
In Class
1-4 p HW
5-21, 24-28
Due Tomorrow

27. Medians and Altitudes of Triangles
Lesson 5.3
Medians and Altitudes
of Triangles

28. Lesson 5.3 Objectives Define a median of a triangle
Identify a centroid of a triangle
Define the altitude of a triangle
Identify the orthocenter of a triangle

29. Triangle Medians
A median of a triangle is a segment that does the following:
Contains one endpoint at a vertex of the triangle, and
Contains the other endpoint at the midpoint of the opposite side of the triangle. A B C D

30. Remember: All medians intersect the midpoint of the opposite side.
Centroid
When all three medians are drawn in, they intersect to form the centroid of a triangle.
This special point of concurrency is the balance point for any evenly distributed triangle.
In Physics, we would call it the center of mass.
Obtuse
Acute
Right

31. Theorem 5.7: 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.
The centroid is 2/3 the distance from any vertex to the opposite side.
AP = 2/3AE
2/3BF
BP = 2/3BF
CP = 2/3CD
2/3AE
2/3CD

32. Example 6
S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the following: RV 6 RU
4 is 2/3 of 6
Divide 4 by 2 and then muliply by 3. Works everytime!! SU 2 RW 12 TS
6 is 2/3 of 9 SV 3

33. Altitudes
An altitude of a triangle is the perpendicular segment from a vertex to the opposite side.
It does not bisect the angle.
It does not bisect the side.
The altitude is often thought of as the height.
While true, there are 3 altitudes in every triangle but only 1 height!

34. Orthocenter
The three altitudes of a triangle intersect at a point that we call the orthocenter of the triangle.
The orthocenter can be located:
inside the triangle
outside the triangle, or
on one side of the triangle
Obtuse
Right
Acute
The orthocenter of a right triangle will always be located at the vertex that forms the right angle.

35. Theorem 5.8: Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a triangle are concurrent.

36. Example 7
Is segment BD a median, altitude, or perpendicular bisector of ABC?
Hint: It could be more than one!
Perpendicular
Bisector
Altitude
None
Median

37. Lesson 5.3 Homework In Class HW Due Tomorrow Quiz Tuesday 1-7p HW
8-23, 39-45
Due Tomorrow
Quiz Tuesday
November 20

38. Midsegment Theorem of Triangles
Lesson 5.4
Midsegment Theorem of
Triangles

39. Lesson 5.4 Objectives Create the midsegment of a triangle
Identify the characteristics of a midsegment of a triangle

40. Midsegment of a Triangle
So far we have studied 4 types of special segments of triangles.
Perpendicular Bisector
Angle Bisector
Median
Altitude
It just so happens that all of these intersect only one side at a time.
And three of the four intersect an vertex and a side.
Another type of special segment is one that connects the midpoints of the sides of a triangle.
This special segment is called the midsegment of a triangle.
Notice there are 3 midsegments in every triangle.

41. Theorem 5.9: Midsegment Theorem of a Triangle
The segment connecting the midpoints of the two sides of a triangle is:
Parallel to the third side
Half the length of the third side
The side it is parallel to
DE = 1/2AC
Segment DE // Segment AC

42. Example 8 Segment MP is the midsegment of LNO. Find x MP = ½NO
P is the midpoint
x = ½(16)
7 = ½(x)
x = 4
x = 8
x = 14

43. Example 9 Fill in the following Segment GJ is parallel to ________.
segment DF
Segment EJ is congruent to _________.
segment JF
Segment DE is parallel to __________.
segment KJ
If EF = 18, then GK = _____. 9
If JK = 13, then ED = _____. 26

44. Lesson 5.4 Homework In Class HW Due Tomorrow 1-11p HW
12-18, 21-29, odds
Due Tomorrow

45. Inequalities in One Triangle
Lesson 5.5
Inequalities in
One Triangle

46. Lesson 5.5 Objectives Compare angle sizes based on side lengths
Utilize the Triangle Inequality Theorem

47. Theorem 5.10: Side Lengths of a Triangle Theorem
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.
Basically, the larger the side, the larger the angle opposite that side.
2nd Largest Angle
Longest side
Smallest Side
Smallest Angle
Largest Angle
2nd Longest Side

48. Theorem 5.11: Angle Measures of a Triangle Theorem
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 angle.
Basically, the larger the angle, the larger the side opposite that angle.
2nd Largest Angle
Longest side
Smallest Side
Smallest Angle
Largest Angle
2nd Longest Side

49. Example 10 Name the smallest and largest angle. Largest Smallest

50. Example 11 Name the smallest and largest side. Largest Smallest

51. Theorem 5.13: Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 3 4 4 3 2 1 6 6 6
Add each combination of two sides to make sure that they are longer than the third remaining side.

52. Example 12
Determine whether the following could be lengths of a triangle.
6, 10, 15
> > > 10 YES!
11, 16, 32
< 32 NO!
Hint: A shortcut is to make sure that the sum of the two smallest sides is bigger than the third side. The other sums will always work.

53. Homework 5.5
In Class
1-5 p HW
6-30 evens
Due End of the Hour