1. PROPERTIES OF TRIANGLES
UNIT FIVE
PROPERTIES OF TRIANGLES

2. 5.1 Perpendiculars and Bisectors

3. 5.1 PERPENDICULAR BISECTORS
Perpendicular Bisector: a segment, ray, line or plane that is perpendicular to a segment at its midpoint C P A B

4. 5.1 PERPENDICULAR BISECTORS
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector, then it is equidistant from the endpoints of the segments.
IF IS A PERPENDICULAR BISECTOR OF THEN C IS EQUIDISTANT FROM A AND B. C A D B

5. 5.1 PERPENDICULAR BISECTORS
Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector. C
If DA = DB, then D lies on the perpendicular bisector of P A B D
D is on

6. 5.1 PERPENDICULARS AND BISECTORS
EXAMPLE 1
In the diagram, MN is
the perpendicular
Bisector of ST.
What segment lengths in the diagram are equal?
Explain why Q is on T 12 Q N M 12
MN bisects ST, so NS = NT. Because M is on the perpendicular bisector of ST, MS=MT (by the perpendicular bisector theorem). The diagram shows that QS=QT=12.
QS=QT, so Q is equidistant from S and T. By the Converse of the Perpendicular Bisector Theorem, Q is on the perpendicular bisector of ST, which is MN. S

7. 5.1 PERPENDICULARS AND BISECTORS
Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. B D A C
If , then DB = DC.

8. 5.1 PERPENDICULARS AND BISECTORS
Converse of Angle Bisector Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. B D A C
If DB = DC, then

9. 5.1 PERPENDICULARS AND BISECTORS
EXAMPLE 2
In the diagram, PM is the bisector of
What is the relationship between
How is the distance between point M and L related to the distance between point M and N? M N L P

10. 5.1 Perpendiculars and Bisectors
Complete the Proof.
Given: D is on the bisector of
Prove: DB = DC B
STATEMENT
REASON 1. 2. 3. 4. 5. 6. D A C

11. 5.2 Bisectors of a Triangle

12. 5.2 BISECTORS OF TRIANGLES
PERPENDICULAR BISECTOR OF A TRIANGLE: a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.
Perpendicular bisector

13. 5.2 Bisectors of Triangles
Concurrent lines: when three or more lines (or rays or segments) intersect in the same point.
Point of Concurrency: the point of intersection of the lines.

14. 5.2 Bisectors of Triangles
The three perpendicular bisectors of a triangle are concurrent.
The point of concurrency is called the circumcenter. P P P
Acute triangle
Right Triangle
Obtuse Triangle

15. 5.2 Bisectors of Triangles
The perpendicular bisectors of a triangle intersect at a point (circumcenter) that is equidistant from the vertices of the triangle. B
PA = PB = PC P A C

16. 5.2 Bisectors of Triangles
Example 1
Three people need to decide on a location to hold a monthly meeting. They will all be coming from different places in the city, and they want to make the meeting location the same distance from each person.
Explain why using the circumcenter as the location for the meeting would be the fairest for all.
Copy the triangle and locate the circumcenter. Tell what segments are congruent.
Since the circumcenter is the same distance from each of the vertices, so each person will have to travel the same distance.
PA = PB = PC C A B

17. 5.2 Bisectors of Triangles
Example 2
The perpendicular bisectors of meet
at point G. Find GC. A E 7 D G C 5 2 F B

18. 5.2 Bisectors of Triangles
Angle Bisector of a Triangle:a bisector of an angle of a triangle.
Incenter of a Triangle: the point of concurrency of the angle bisectors.
Incenter

19. 5.2 Bisectors of Triangles
Concurrency of Angle Bisectors of a Triangle: The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. B
PD = PE = PF D F P A C E

20. 5.2 Bisectors of Triangles
Example 3
The angle bisectors of meet at point M. Find MK. L X Z 8 12 5 K J Y

21. 5.3 Medians and Altitudes of a Triangle

22. 5.3 Medians and Altitudes
Median of a Triangle: a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Centroid: the point of concurrency of the medians
OBUSE TRIANGLE
ACUTE TRIANGLE
RIGHT TRIANGLE

23. 5.3 Medians and Altitudes
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. B D P E C F A

24. 5.3 Medians and Altitudes Example 1
P is the centroid of shown below and PT = 5. Find RT and RP. R
RP = 10, RT = 15 P S Q T

25. 5.3 Medians and Altitudes Example 2
Find the coordinates of the centroid of
(3, 4)

26. 5.3 Medians and Altitudes
Altitude of a triangle: the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.
Orthocenter: the point of concurrency of the altitudes. The orthocenter can lie inside, outside, or on the triangle.
Orthocenter

27. 5.3 Medians and Altitudes Example 3
Where is the orthocenter located in If
Inside on
outside

28. 5.3 Medians and Altitudes
Example 4: Use the diagram to match the type of special segment with the correct segment. Z T Y U X W V
Median A. ZX
Altitude B. ZW
Perpendicular Bisector C. ZV
Angle Bisector D. TV

29. 5.4 Midsegment Theorem

30. 5.4 Midsegment Theorem
Midsegment of a Triangle: a segment that connects the midpoints of two sides of a triangle. B
If D is the midpoint of AB and E is the midpoint of BC, then DE is a midsegment. D E A C

31. 5.4 Midsegment Theorem Example 1
Show that the midsegment MN is parallel to side JK and is half as long.
K (4, 5)
J (-2, 3) N M
L(6, -1)

32. 5.4 Midsegment Theorem
Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. C D E A B

33. 5.4 Midsegment Theorem Example 2
and are midsegments of Find UW and RT. R U 12 V
RT = 16, UW = 6 8 T W S

34. 5.4 Midsegment Theorem Example 3 What are the coordinates of Q and R?
Why is QR MP?
What is MP? What is QR?
(a/2, b/2); (a+c/2, b/2)
Both have a slope of zero.
c. c; c/2
N (a, b) Q R
P (c, 0)
M (0, 0)

35. 5.4 Midsegment Theorem Example 4
The midpoints of the sides of a triangle are A(2, 5), B(2, 2), and C(6, 5).
What are the coordinates of the vertices of the triangle?
Find the perimeter of the triangle.
(6, 2) (6, 8) and (-2, 2)
24 units

36. 5.5 Inequalities in One Triangle

37. 5.5 Inequalities in One Triangle
Triangle Theorems
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. B 5 3 C A

38. 5.5 Inequalities in One Triangle
Triangle Theorems (continued)
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. F 40 D 60 E

39. 5.5 Inequalities in One Triangle
Example 1: Write the measurements of the triangles in order from least to greatest.
a b. P J
100 5 8 35 45 H G Q R 7

40. 5.5 Inequalities in One Triangle
Exterior Angle Inequality: The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. A 1 C B

41. 5.5 Inequalities in One Triangle
Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A C B

42. 5.5 Inequalities in One Triangle
Example 2
Given the possible triangle side lengths, which groups could form a triangle?
2cm, 2cm, 5cm b. 3cm, 2cm, 5cm
c. 4cm, 2cm, 5cm c

43. 5.5 Inequalities in One Triangle
Example 3
A triangle has one side of 10 cm and another of 14 cm. Describe the possible lengths of the third side.
4>x>24

44. 5.5 Indirect Proof and Inequalities in Two Triangles

45. 5.6 Indirect Proof and Inequalities in Two Triangles
Indirect Proof: A proof in which you prove that a statement is true by first assuming that its opposite is true.
If this assumption leads to an impossibility, then you have prove that the original statement is true.

46. Prove: does not have more than one obtuse angle.
EXAMPLE 1: Given:
Prove: does not have more than one obtuse angle.
Assume that has more than one obtuse angle.
2. You know, however, that the sum of the measures of all three angles is 180. 1. 2.
Assume that ABC has two obtuse angles.
Add the two given inequalities.
Triangle Sum Theorem
Subtraction Property of Equality 3. 4.

47. So, you can substitute for
The last statement is not possible; angle measures in triangles cannot be negative.
So, you can conclude that the original assumption is false. That is, triangle ABC cannot have more than one obtuse angle. 5. 6.
Substitution Property of Equality
Simplify

48. 5.6 Indirect Proof and Inequalities in Two Triangles
Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. V R
RT > VX
100 80 S T W X

49. 5.6 Indirect Proof and Inequalities in Two Triangles
Converse of the Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. B E 7 8 D C F A

50. 5.6 Indirect Proof and Inequalities in Two Triangles
Example 2: Complete each with <, > or =.
a b c. K E D 38 26 47 L M 2 27 1 N 37 F C 45 Q P

51. 5.6 Indirect Proof and Inequalities in Two Triangles
Example 3: You and a friend are flying separate planes. You leave the airport and fly 120 miles due west. You then change direction and fly W 30 N for 70 miles. Your friend leaves the airport and flies 120 miles due east. She then changes direction and flies E 40 S for 70 miles. Each of you has flown 190 miles, but which plane is farther from the airport?