1. Ch 5 Goals and Common Core Standards Ms. Helgeson
Use properties of perpendicular bisectors.
Use properties of angle bisectors to identify equal distances.
Use properties of perpendicular bisectors of a triangle.
Use properties of angle bisectors of a triangle.
Use properties of medians and altitudes of a triangle.
Identify the midsegments of a triangles.

2. Use properties of midsegments of a triangle.
Use triangle measurements to decide which side is longest or which angle is largest.
Use the triangle Inequality.
Use the Jhinge theorem and its converse to compare side lengths and angle measures.
CC.9-12.G.GPE.2
CC.9-12.G.C.3

3. CC.9-12.G.GPE.4
CC.9-12.G.CO.10
CC.9-12.G.SRT.8

4. Relationships in Triangles
TOPIC 5
Relationships in Triangles

5. 5.1 Perpendicular and Angle Bisectors
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. C P B A
If CP is the perp. bisector Of AB, then CA = CB.

6. Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
If DA = DB, then D lies on the perp. Bisector of AB. C P B A D

7. What is the value of AD? A
3x + 2
6x - 10 B D 11 C 11

8. What is the value of OL? K 17 17 O L J 14 N 9 9 M

9. The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. Q m P

10. Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
If m<BAD = m<CAD, then DB = DC B D A C

11. Converse of the 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.
If DB = DC, then m<BAD = m<CAD B D A C

12. If HI = 7, IJ = 7, and m <HGI = 25˚, what is m <IGJ
If HI = 7, IJ = 7, and m <HGI = 25˚, what is m <IGJ? If m <HGJ = 57˚, m <IGJ = 28.5˚, and HI = 12.2, what is the value of IJ? H I G J

14. 5.2 Bisectors of a Triangle
A perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.
Perpendicular bisector

15. Draw a right triangle, acute triangle, and obtuse triangle
Draw a right triangle, acute triangle, and obtuse triangle. Find the perpendicular bisector for all the sides of the triangle. The perpendicular bisectors should meet at one point (circumcenter). (Acute-inside; obtuse-outside; right-on hypotenuse) Measure the distance from the circumcenter to each vertex of the triangle. Distances are the same. Draw the circumscribed circle.

16. The three perp. Bisectors of a triangle are concurrent
The three perp. Bisectors of a triangle are concurrent. The point of concurrency can be inside the triangle, on the triangle, or outside the triangle. B
The perp. Bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. P
PA = PB = PC C A
Page 272
The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. Page 273

17. An angle bisector of a triangle is a bisector of an angle of the triangle. The three angle bisectors are concurrent.
Angle bisectors P
The point of concurrency of the angle bisectors is called the incenter of the triangle. Page 274

18. Draw a triangle (acute, obtuse, right)
Draw a triangle (acute, obtuse, right). Find the angle bisectors of each angle. The angle bisectors should me at one point (incenter). Measure the distance from incenter to each side. (distance-perpendicular distance to side. All the distances are the same. Draw the inscribed circle.

19. B
The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. F D P
PD = PE = PF C A E
The point of concurrency of the angle bisectors is called the incenter of the triangle. Page 274

20. If m <BAF = 32˚ and m <CBF = 41˚, what is m <ACF
If m <BAF = 32˚ and m <CBF = 41˚, what is m <ACF? If EF = 3y – 5 and DF = 2y + 4, what is the distance from F to AB? A E C F D B

22. 5.3 The median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Median

23. Draw a triangle. Construct the three medians
Draw a triangle. Construct the three medians. The medians meet at one point (centroid and is the center of gravity for the triangle). If you cut out triangle and make an indentation with pencil through the centroid, the triangle should balance. If not done correctly, the triangle will tilt to one side and fall off.

24. Coordinates of centriod = (average of x coordinates, average of y coordinates)P A
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.
AP = 2/3 AD, etc
Concurrency of Medians of Triangle Theorem
The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle.

25. An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle.
altitude

26. The lines containing the altitudes of a triangle are concurrent.
The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle. Acute triangle-orthocenter inside triangle, obtuse triangle – outside triangle, and right triangle, on triangle.

27. Activity 1: Triangle ABC has vertices A(-3, 10), B(9, 20), and C(-2, 21). Find the coordinates of P such that CP is a median of triangle ABC. Determine if CP is an altitude of triangle ABC.

28. Activity 2: The following equations intersect to form a triangle
Activity 2: The following equations intersect to form a triangle. x – 2y = -6 3x + 2y = -2 9x – 2y = 26 Identify the vertices of the triangle. Draw one of the perpendicular bisectors in the triangle and identify the slope and point used to draw it. Write the equation for that perpendicular bisector.

29. Activity 3: Draw a scalene triangle ABC
Activity 3: Draw a scalene triangle ABC. Find the midpoint of each side by folding each side, vertex to vertex, and pinching the paper in the middle. The midpoint of AB should be E, the midpoint of BC should be D, and the midpoint of AC should be F. The centroid should be labeled as P. Draw the medians from the vertices to the midpoints. next slide

30. Measure the lengths of the segments in the table below and record the measurements. Does there seem to be a relationship between the distance from P to a vertex and the distance from that vertex to the midpoint of the opposite side?
Length of Median
AD = ?
BF = ?
CE = ?
Length of segment from P to vertex
AP = ?
BP = ?
CP = ?
Median length multiplied by 2/3
2/3(AD) = ?
2/3(BF) = ?
2/3(CE) = ?

31. Find the ratios: AP/PD = BP/PF = CP/PE =

32. Activity: Plot the points H(-9, 3), A(-3, 7), and T(3, 5) and draw triangle HAT. Find the midpoints of each side and label the midpoint of HA as G, and midpoint of HT as D, and midpoint of AT as O. Do not draw medians at this time. Find the point that divides HO into a ratio of 2:1, AD into a ratio of 2:1, TG into a ratio of 2:1. Plot point and label it C. next slide

33. What does this point represent. Draw the medians
What does this point represent? Draw the medians. Verify by observation that the three medians pass through C. Verify that the centroid is located two-thirds of the distance from the vertex to the midpoint of the opposite side using the distance formula.

34. 5.4 Midsegment Theorem and Inequalities in One Triangle
Definition:
A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.
Midsegment AB
Midpoint Formula
X X 2 , Y1 + Y2 A B

35. Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle is
A) parallel to the third side, and
B) is half as long. C
DE AB AND DE = ½ AB D E B A

36. Inequalities in One Triangle
Shortest
side
Largest angle
Longest side
Smallest
angle

37. 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.
Converse of the above theorem
If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer then the side opposite the smaller angle.

38. Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.
M<1 > m<A and m<1 > m< B A 1 C B

39. Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
AB + BC > AC
AC + BC > AB
AB + AC > BC A B C

40. Finding possible side lengths
A triangle has one side of 10 cm. and another of 14 cm. Describe the possible lengths of the third side.
Find the value of x and list the length of the sides of triangle ABC in order from shortest to longest if m<A = 5x – 5, m<B = 4x, and m<C = 17x + 3.

41. Triangle DEF has vertices D(-2, 1), E(5, 3), and F(4, -3)
Triangle DEF has vertices D(-2, 1), E(5, 3), and F(4, -3). List the angles in order from greatest to least.

42. 5.5 Inequalities in Two Triangles

43. 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.
BC > GH

44. Converse of the Hinge Theorem
If two sides of 1 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.

45. Ex: What are the possible values of x?D 18
(2x – 4)˚ C G A
36˚
60˚ E 14 21 F B