More Triangles.

1
Chapter 5

• More Triangles.
• Mr. Thompson

More Triangles. Mr. Thompson.
2

• Midsegment Theorem

3

• A midsegment of a triangle is a segment that
  connects the midpoints of
• two sides of
• a triangle.

4
Midsegment Theorem

• The segment connecting the midpoints of 2 sides
  of a triangle is parallel to
• the 3rd side
• and is ½ as
• long.

5

• Perpendiculars and Bisectors

6

• In 1.5, you learned that a segment bisector
  intersects a segment at its midpoint.

midpoint
10
10
Segment bisector
7
Perpendicular bisector

• A segment, ray, line, or plane that is
  perpendicular to a segment at its midpoint is
  called a perpendicular bisector.

d
f
12
12
8
Y is equidistant from X and Z.

• A point is equidistant from two points if its
  distance from each point is the same.

x
z
y
9
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
8
A
B
10
Converse of the Perpendicular Bisector Theorem

• If a point is equidistant from the endpoints of
  a segment,

x
z
y
Y is equidistant from X and Z.
11
Using Perpendicular Bisectors
What segment lengths in the diagram are equal?
T
12
NSNT (given) M is on the perpendicular bisector
of ST, so.. MSMT (Theorem 5.1) QS QT12 (given)
Q
N
M
12
S
12
Using Perpendicular Bisectors
Explain why Q is on MN.
QSQT, so Q is equidistant from S and T.
T
12
Q
N
M
By Theorem 5.2, Q is on the perpendicular
bisector of ST, which is MN.
12
S
13
The distance from a point to a line..

• defined as the length of the perpendicular
  segment from the point to the line.

R
The distance from point R to line m is the length
of RS.
m
S
14
Point that is equidistant from two lines

• When a point is the same distance from one line
  as it is from another line, the point is
  equidistant from the two lines(or rays or
  segments).

15
Angle Bisector Theorem

• If a point is on the bisector of an angle, then
  it is equidistant from the 2 sides of the angle.
• If angle ABD angle CBD,
• then DC AD.

16
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 DC AD, then angle ABD angle CBD.

17
Bisectors of a Triangle
18

• 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.

19
Investigation

• of the Perpendicular Bisector Theorem.

20

• When three or more lines (or rays or segments)
  intersect in the same point, they are called
  concurrent lines (or rays or segments). The
  point of
• intersection of the
• lines is called the
• point of
• concurrency.

21

• The three perpendicular bisectors of a triangle
  are concurrent. The point of concurrency can be
  inside the triangle, on the triangle, or outside
  the triangle.

22

• The point of concurrency of the perpendicular
  bisectors
• of a triangle is
• called the
• circumcenter
• of the triangle.

23
Concurrency of Perpendicular Bisectors of a
Triangle

• The perpendicular bisectors of a triangle
  intersect at a point
• that is equidistant
• from the vertices
• of the triangle.
• OA1 OA2 OA3

24

• An angle bisector of a triangle is a bisector of
  an angle of the triangle. The three angle
  bisectors are concurrent. The point of
  concurrency of the angle bisectors is called the
• incenter of the
• triangle and is
• always inside the
• triangle.

25
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. LMC LMA LMB
26
Classwork
Page 246 6, 13, 31, 32, 35, 38 Page 252 28,
29, 33, 40, 46
27

• Medians and Altitudes of a Triangle

28
Medians and Altitudes
29

• A median of a triangle is a segment whose
  endpoints are a vertex of the
• triangle and the
• midpoint of the
• opposite side.

30
The three medians of a triangle are concurrent.
The point of concurrency is called the centroid
of the triangle. The centroid is always inside
the triangle.
31
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.

32

• 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.

33

• If AR, CT, and BU are altitudes of triangle ABC,
  then AR, CT,
• and BU intersect
• at some point P.

34
Example

• ?ABC A(-3,10), B(9,2), and C(9,15)
• a) Determine the coordinates of point P on AB
  so that CP is a median of ?ABC.
• b) Determine if CP is an altitude of ?ABC

35
Example
2) ?SGB S(4,7), G(6,2), and B(12,-1) a)
Determine the coordinates of point J on GB so
that SJ is a median of ?SGB b) Point M(8,3).
Is GM an altitude of ?SGB ?
36

• Inequalities in One Triangle

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.

38

• 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
• side.

39
Triangle Inequality

• The sum of the lengths of any two sides of a
  triangle is greater than the length of the third
  side.
• Would sides of length 4, 5 and 6 form a
  triangle....?
• How about sides of length 4, 11, and 7 ?