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More About Triangles

§ 6.1 Medians

§ 6.2 Altitudes and Perpendicular Bisectors

§ 6.3 Angle Bisectors of Triangles

§ 6.4 Isosceles Triangles

§ 6.5 Right Triangles

§ 6.6 The Pythagorean Theorem

§ 6.7 Distance on the Coordinate Plane
Medians
You will learn to identify and construct medians in triangles
1) ______
median
2) _______
centroid
3) _________
concurrent
Medians
vertex of the triangle and
In a triangle, a median is a segment that joins a ______
midpoint of the side __________________.
opposite that vertex
the ________
A
median CF
F
E
median BE
B
D
C
median AD
The medians of ΔABC, AD, BE, and CF, intersect at a common point
centroid
called the ________.
When three or more lines or segments meet at the same point, the lines are
concurrent
__________.
Medians
There is a special relationship between the length of the segment from the
vertex to the centroid and the length of the segment from the centroid to the
midpoint.
A
F
E
B
D
C
Medians
The length of the segment from the vertex to the centroid is
_____
twice the length of the segment from the centroid to the
midpoint.
Theorem 6 - 1
2x
x
When three or more lines or segments meet at the same point,
concurrent
the lines are __________.
Medians
AD, BE, and CF are medians of ABC.
What is the measure of CD if CE  4 x  3,
DB  5 x  1, and EA  2 x  9 ?
A
CD = 14
F
E
B
D
C
Altitudes and Perpendicular Bisectors
You will learn to identify and construct _______
altitudes and
__________________
perpendicular bisectors in triangles.
1) ______
altitude
2) __________________
perpendicular bisector
Altitudes and Perpendicular Bisectors
In geometry, an altitude of a triangle is a ____________
perpendicular segment with one
vertex and the other endpoint on the side opposite
endpoint at a ______
_______ that
vertex.
A
B
D
C
The altitude AD is perpendicular to side BC.
Altitudes and Perpendicular Bisectors
Constructing an altitude of a triangle
4)
straightedge
align
1) Use
Drawaathe
triangle
like to
ΔABC
3)
Place
compass
point
at
DB and
2)
onthe
and
vertex
B arc
andbelow
the intersects
point
the
draw an
AC.where
Using
the
that
side
AC
two
arcs
intersect.
Draw
a segment
same
compass
setting,
the
in two
points.
Label
theplace
points
of
from
the vertex
B to
AC. Label
compass
point
on
EE.side
and draw
an
intersection
D and
the
point
of intersection
F.
arc to
intersect
the one drawn.
A
B
D
E
C
Altitudes and Perpendicular Bisectors
An altitude of a triangle may not always lie inside the triangle.
Altitudes of Triangles
acute triangle
right triangle
obtuse triangle
The altitude is
inside the triangle
_____
The altitude is
a side of the triangle
_____
The altitude is
out side the triangle
_______
Altitudes and Perpendicular Bisectors
Another special line in a triangle is a perpendicular bisector.
side of a triangle is called the
A perpendicular line or segment that bisects a ____
perpendicular bisector of that side.
Line m is the perpendicular bisector
of side BC.
B
m
A
altitude
D
D is the midpoint of BC.
C
Altitudes and Perpendicular Bisectors
In some triangles, the perpendicular bisector and the altitude are the same.
Y
The line containing YE is the
perpendicular bisector of XZ.
YE is an altitude.
X
E is the midpoint of XZ.
E
Z
Angle Bisectors of Triangles
You will learn to identify and use ____________
angle bisectors in triangles.
1) ___________
angle bisector
Angle Bisectors of Triangles
Recall that the bisector of an angle is a ray that separates the angle into two
congruent angles.
Q
P
S
R
PS bisects QPR
QPS  SPR
mQPS  mSPR
Angle Bisectors of Triangles
An angle bisector of a triangle is a segment that separates an angle of the
triangle into two congruent angles.
vertex of the triangle,
One of the endpoints of an angle bisector is a ______
opposite that vertex.
and the other endpoint is on the side ________
B
C
D
A
AC is an angle bisector of DAB
DAC  CAB
mDAC  mCAB
Just as every triangle has three medians, three altitudes, and three
perpendicular bisectors, every triangle has three angle bisectors.
Angle Bisectors of Triangles
Special Segments in Triangles
Segment
 altitude
 perpendicular
bisector
 angle bisector
Type
 line segment
 line
 ray
 line segment
 line segment
bisects the side
of the triangle
bisects the angle
of the triangle
Property
from the vertex, a
line perpendicular
to the opposite
side
Isosceles Triangles
You will learn to identify and use properties of _______
isosceles
triangles.
1) isosceles
_____________
triangle
2) ____
base
3) ____
legs
Isosceles Triangles
Recall from §5-1 that an isosceles triangle has at least two congruent sides.
legs
The congruent sides are called ____.
The side opposite the vertex angle is called the base
____.
In an isosceles triangle, there are two base angles, the vertices where the
base intersects the congruent sides.
vertex angle
leg
leg
base angle
base
base angle
Isosceles Triangles
Theorem
6-2
Isosceles
Triangle
Theorem
6-3
If two sides of a
triangle are congruent,
then the angles
opposite those sides
are congruent.
The median from the
vertex angle of an
isosceles triangle lies
on the perpendicular
bisector of the base
and the angle bisector
of the vertex angle.
A
If AB  AC , then
C  B
B
C
A
If AB  AC and
BD  CD, then
B
D
C
AD  BC and
BAD  CAD
Isosceles Triangles
Theorem
6-4
Converse of
Isosceles
Triangle
Theorem
If two angles of a
triangle are congruent,
then the sides opposite
those angles
are congruent.
A
If B  C , then
AB  AC
B
C
A triangle is equilateral if and only if it is equiangular.
Theorem
6-5
Right Triangles
You will learn to use tests for _________
congruence of ____
right triangles.
1) _________
hypotenuse
2) ____
legs
Right Triangles
In a right triangle, the side opposite the right angle is called the
hypotenuse
_________.
legs
The two sides that form the right angle are called the ____.
leg
leg
Right Triangles
Recall from Chapter 5, we studied various ways to prove triangles to be
congruent:
In §5-5, we studied two theorems
S
B
C
A
T
R
and
S
B
A
C
R
T
Right Triangles
Recall from Chapter 5, we studied various ways to prove triangles to be
congruent:
In §5-6, we studied two theorems
S
B
C
A
T
R
and
S
B
A
C
R
T
Right Triangles
The theorems mentioned in Chapter 5, were for ALL triangles.
So, it should make perfect sense that they would apply to right triangles
as well.
Theorem
6-6
LL Theorem
If two legs of one right triangle are congruent to the
corresponding legs of another right triangle, then the triangles
are congruent.
D
A
B
C
E
same as
ABC  DEF
F
Right Triangles
The theorems mentioned in Chapter 5, were for ALL triangles.
So, it should make perfect sense that they would apply to right triangles
as well.
Theorem
6-7
HA Theorem
If ______________
the hypotenuse and an (either) __________
acute angle of one right
triangle are congruent to the __________
hypotenuse and
_________________
corresponding angle of another right angle, then the
triangles are congruent.
D
A
B
C
E
same as
ABC  DEF
F
Right Triangles
The theorems mentioned in Chapter 5, were for ALL triangles.
So, it should make perfect sense that they would apply to right triangles
as well.
Theorem
6-6
LA Theorem
If one (either) ___
acute angle of a right triangle are
leg and an __________
congruent to the ________________________
corresponding leg and angle of another
right triangle, then the triangles are congruent.
D
A
B
C
E
same as
ABC  DEF
F
Right Triangles
The theorems mentioned in Chapter 5, were for ALL triangles.
So, it should make perfect sense that they would apply to right triangles
as well.
Postulate
6-1
HL Postulate
If the hypotenuse and a leg on one right triangle are
congruent to the hypotenuse and corresponding leg of
another right triangle, then the triangles are congruent.
B
Theorem?
D
A
C
E
ABC  DEF
F
Pythagorean Theorem
You will learn to use the __________
Pythagorean Theorem and its converse.
1) _________________
Pythagorean Theorem
2) _______________
Pythagorean triple
* 3) converse
_______
Pythagorean Theorem
If two
___ measures of the sides of a _____
right triangle are known, the
side
___________________
Pythagorean Theorem can be used to find the measure of the third ____.
c
a
c  a b
2
2
2
c  a 2  b2
b
Pythagorean triple is a group of three whole numbers that satisfies
A _________________
the equation c2 = a2 + b2, where c is the measure of the hypotenuse.
5
3
4
52 = 32 + 42
25 = 9 + 16
Pythagorean Theorem
Theorem
6-9
Pythagorean
Theorem
In a right triangle, the square of the length of the
c is equal to the sum of the squares of the
hypotenuse __,
b
a and __.
lengths of the legs __
c
a
c2  a2  b2
b
Theorem 6-10 If c is the measure of the longest side of a triangle, a and b
Converse of the are the lengths of the other two sides, and c2 = a2 + b2,
Pythagorean
Theorem
then the triangle is a right angle.
Distance on the Coordinate Plane
You will learn to find the ______________________on
the
distance between two points
coordinate plane.
Nothing new!
You learned this in Algebra I.
d
x2  x1    y2  y1 
2
2
Distance on the Coordinate Plane
y
1) On grid paper, graph A(-3, 1) and C(2, 3).
2) Draw a horizontal segment from A
and a vertical segment from C.
C(2, 3)
A(-3, 1)
B(2, 1)
3) Label the intersection B and find
the coordinates of B.
QUESTIONS:
What is the measure of the distance between A and B?
(x2 – x1) = 5
What is the measure of the distance between B and C?
(y2 – y1) = 2
What kind of triangle is ΔABC?
right triangle
If AB and BC are known, what theorem can be used to find AC?
Pythagorean Theorem
What is the measure of AC?
29 ≈ 5.4
x
Distance on the Coordinate Plane
If d is the measure of the distance between two points with
coordinates (x1, y1) and (x2, y2),
Theorem
6-11
Distance
Formula
y
B(x2, y2)
d
A(x1, y1)
then d =
x
x2  x1 2   y2  y1 2
Find the distance between each pair of points.
Round to the nearest tenth, if necessary.
M 2, 3, N5, 7
5
T 6, 4, U2, 2
4.5
Distance on the Coordinate Plane