Midsegments of Triangles

1
Section 5-1

• Midsegments of Triangles

2
Definition of Midsegment

• The midsegment of a triangle is a segment
  connecting the midpoints of 2 of the sides.

3
Theorem 5-1Triangle Midsegment Thm.

• If a segment joins the midpoints of 2 sides of a
  triangle, then the segment is parallel to the
  third side, and is half its length.
• _at_

4
Identifying Parallel Segments
5
Identifying Lengths of Segments
AC? BC? AB?
25
20
20
6
Using Algebra!

• Solve for x if
• XY x2 and
• BC 46.

x2
7
More Algebra

• Find x if BC 5x 16.

12
8
Try another one!

• Solve for x if
• XZ 5x - 4 and
• AC 36.

5x - 4
9
Closure

• The perimeter of a triangle is 78 ft. Find the
  perimeter of the triangle formed by its
  midsegments.

10
Homework

• P.246
• A) 2, 6, 8, 12-14, 22, 24, 32
• B) 4, 10, 11, 16, 18, 25, 30, 35
• C) 26, 38, 39 (Pick two)

11
Section 5-2

• Bisectors of Triangles

12
Theorem 5-2Perpendicular Bisector Theorem

• If a point is on the perpendicular bisector of a
  segment, then it is equidistant from the
  endpoints of the segment.

13
Theorem 5-3Converse of the Perpendicular
Bisector Theorem

• If a point is equidistant from the endpoints of
  the segment, then it is on the perpendicular
  bisector of a segment.

14
Theorem 5-4Angle Bisector Theorem

• If a point is on the bisector of an angle, then
  the point is equidistant from the sides of the
  angle.

15
Theorem 5-5Converse of the Angle Bisector Theorem

• If a point is equidistant from the sides of an
  angle, then the point is on the bisector of the
  angle.

16

• Find the value of x.

5x
2x 24
Find FD and FB
17
How far is K from EH?
How far is K from ED?
D

• Find the value of x

2x
K
E
C
10
x 20
What can you conclude about EK?
H
What is the measure of ?DEH?
What is the measure of ?DKE?
18
Closure

• State the perpendicular bisector theorem and its
  converse as a biconditional

A point is on the perpendicular bisector of a
segment if and only if it is equidistant from the
endpoints of the segment
19
Homework

• Page 251
• A) 2,6,8,12,13,14,22,24,32
• B) 4,10,11,16,18,25,30,35 Pick 4
• C) 26,38,39 pick 2

20
Section 5-3

• Concurrent Lines, Medians, and Altitudes

21
Vocabulary

• Concurrent When 3 or more lines all intersect
  in one point, they are concurrent.
• Point of Concurrency The point at which the
  lines intersect.

22
Theorem 5-6

• The perpendicular bisectors of the sides of a
  triangle are concurrent at a point equidistant
  from the vertices of the triangle.
• This point is called the CIRCUMCENTER.
• This point is the center of a circle that is
  circumscribed about the triangle.

23
Circumcenter
AO BO CO
24
Theorem 5-7

• The bisectors of the angles of a triangle are
  concurrent at a point equidistant from the sides
  of the triangle.
• This point is called the INCENTER.
• This point is the center of a circle that is
  inscribed in the circle.

25
Incenter
26
Theorem 5-8

• The medians of a triangle are concurrent at a
  point that is 2/3 the distance from each vertex
  to the midpoint of the opposite side.
• This point is called the CENTROID.
• This point is the center of gravity for the
  triangle.

27
Centroid
GM 2/3 AG
28
Theorem 5-9

• The lines that contain the altitudes of a
  triangle are concurrent.
• This point is called the ORTHOCENTER.

29
Orthocenter
30
Cool, huh?
EULER LINE
31
Closure

• Where is the center of a circumscribed circle on
  a right angle???

On the hypotenuse
32
Homework

• Page 259
• A) 2,4,8,12,14,16,20,22,28
• B) 6,13,15,19,21,27,29,42 pick 4
• C) 32,35 pick 1

33
Section 5-4

• Inverses, Contrapositives and Indirect Reasoning

34
Reviewing conditionals

• If a triangle has two equal sides, then it is an
  isosceles triangle.
• What is the converse?
• Can you write a biconditional?

35
Negating Statements

• The negation of a statement has the opposite
  truth value.
• Example
• The statement p says A polygon has 4 sides
• The negation of p says A polygon does not have 4
  sides
• One is true while the other is false.
• Symbolism for not p is p

36
Inverses

• The inverse of a conditional negates both the
  hypothesis and the conclusion.
• The conditional is p ? q so the inverse is
  defined as p ? q. If not p, then not q.
• Example
• Conditional If a figure is a square, then it is
  a rectangle.
• Inverse If a figure is not a square, then it is
  not a rectangle.

37
Contrapositives

• A contrapositive of a conditional switches the
  hypothesis and the conclusion and negates them
  both.
• The contrapositive is q ? p. If not q, then
  not p.
• If a figure is not a rectangle, then it is not a
  square.

38
Equivalent Statements

• Equivalent statements have the same truth value.
• A conditional and its contrapositive are always
  equivalent statements.
• Can you make up an example to show this?

39
Writing an indirect proof

• Step 1 State as an assumption the opposite
  (negation) of what you want to prove.
• Step 2 Show that this assumption leads to a
  contradiction.
• Conclude that the assumption must have been false
  and therefore what you want to prove must be true.

40
Closure

• Determine which are true the conditional below,
  its converse, its inverse, or it contrapositve

If lines are perpendicular, then they are not skew
Conditional and contrapositive
41
Homework

• Page 267
• A) 2,6,8,10,12,16,17,21,22,31,34
• B) 4,9,14,18,23,32,35 pick 4
• C) 40,41 pick 1

42
Section 5-5

• Inequalities in Triangles

43
Comparison Property of Inequality

• If a b c and c gt 0, then agtb.
• Proof

Statements Reasons

• c gt 0 1. Given
• bcgtb0 2. Add. Prop.
• bcgtb 3. Simplify
• abc 4. Given
• agtb 5. Subst. a for bc in 3

44
Corollary to the Exterior Angle Theorem

• The measure of an exterior angle of a triangle is
  greater than the measure of either of its remote
  interior angles.
• mlt1 is greater than mlt2 and the mlt3.

45
Theorem 5-10

• If 2 sides of a triangle are not congruent, then
  the larger angle lies opposite the longer side.
• If ZX gt ZY then angle Y is greater than angle X.

46
Theorem 5-11

• If 2 angles of a triangle are not congruent, then
  the longer side is opposite the larger angle.
• If angle X is greater than angle Y, then ZYgtZX.

47
Theorem 5-12Triangle Inequality Theorem

• The sum of the lengths any 2 sides of a triangle
  is greater than the length of the third side.
• XY YZ gt XZ
• XY XZ gt YZ
• YZ XZ gt XY

48
Closure

• Explain why each triangle below is impossible

12
15
18
12
1000
30
50
25
32
In the first triangle, the side opposite the
smallest angle is not the shortest side the
second triangle violates the Triangle Inequality
Theorem
49
Homework

• Page 276
• A) 2,4,10,14,16,20,22,32,34
• B) 3,6,12,18,24,26,36 pick 4
• C) 28,38,39 pick 2