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Segment Length and Midpoints
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by
TweetJuan Jorrin
on 22 November 2018Transcript of Segment Length and Midpoints
Distance Formula
Midpoint Formula
Segment Addition Postulate
Defined Terms
The Three Undefined Terms
Segment Length and Midpoints
POint
Line
Line Segment
Ray
Other Terms
they served as the foundation of geometry that are used to define other terms
the most basic terms in Geometry that don’t require formal definitions
has no size or dimension
has location
M
represented by a dot
named using a capital letter
is a collection of points that extends infinitely in both directions
has no thickness
named using a small letter or by any two points on the line
n
X
Y
line
n
XY
YX
has two dimensions, the length and the width
has no height
flat surface that extends infinitely in both directions
O
A
C
B
named using a letter or by 3 points on the plane
plane
O
plane
ABC
point
M
terms with formal definition
part of a line
A
B
consists of two endpoints and all the points in between them
named using the two endpoints
segment
AB
segment
BA
AB
BA
C
D
part of a line that begins with an endpoint and ends infinitely
named using two points, the first point must be the endpoint
ray
CD
CD
Coplanar Points
Collinear Points
Parallel Lines
Postulate
points that are on the same plane
points that are on the same line
lines on the same plane that will not intersect
statement that is considered true without proof
X
Z
Y
XY
YZ
XZ
states that if we are given a line segment with endpoints
XY
X
and
Z
,
there is a point Y lying between points
X
and
Z
if and only if
+ YZ
= XZ
1. Use the segment addition postulate to answer this problem.
A
B
C
3
x
 6
6
x
+ 4
Given that AC=79,
find AB and BC.
STEP 1.
Find x using the segment addition postulate.
AB + BC = AC
Substitute A=3x6, B=6x+4 C=79.
(
3
x
 6
) + (
6
x
+ 4
) =
79
Add like terms.
9
x
 2 = 79
Add both sides by 2.
9
x
 2
+ 2
= 79
+ 2
9
x
= 81
Divide both sides by 9.
9
9
x
= 9
STEP 2.
Use
x
= 9 to find
AB
.
AB
= 3
x
 6
= 3(
9
)  6
= 27  6
AB
= 21
Substitute x=9.
Simplify.
STEP 3.
Use
x
= 9 to find
BC
.
BC
= 6
x
+ 4
= 6(
9
) + 4
= 54 + 4
BC
= 58
Substitute x=9.
Simplify.
STEP 3.
(Optional) Check the answers.
Use the segment addition postulate.
AB
+
BC
=
AC
21
+
58
=
79
79 = 79
Substitute the given AC=79 and our answers B=21 and BC=58. Simplify.
The statement is TRUE, then ours answers are correct!
D
= √(
x  x
) + (
y  y
)
2
1
2
2
1
2
The distance between two points
P
(
x
,
y
) and
P
(
x
,
y
) is given below
1
1
1
2
2
2
2. Determine the length of the line segment AB.
0
1
3
5
2
y
x
A
(1,3)
B
(5,2)
AB
= √(
x

x
) + (
y

y
)
1
2
1
2
2
2
= √(
5

1
) + (
2

3
)
= √4 + (5)
= √16 + 25
= √41
M
x + x
2
1
2
y + y
2
1
2
,
The midpoint
M
of a line segment with endpoint
A
(
x
,
y
) and
B
(
x
,
y
) can be found using the formula below
1
2
1
2
A
B
M
x
y
4. Find the midpoint of the two points (5, 4) and (5, 2).
5. Find x and y if (6, 5) is the midpoint of points (x, y) and (4, 7).
3. Find
x
so that the distance between the points (2, 3) and (
x
,5) is equal to 10.
2
2
2
2
Use the distance formula.
Identify (x , y ) and (x , y ). Plug the values in the formula.
1
1
2
2
(
x
,
y
)
1
1
(
x
,
y
)
2
2
Simplify.
Thus, the length of line segment AB is √41.
D
= √(
x

x
) + (
y

y
)
1
2
1
2
2
2
10
= √(
x

2
) + (
5
 (
3
))
2
2
Use the distance formula.
Identify (x , y ), (x , y ) and D. Plug the values in the formula.
1
1
2
2
(
x
,
y
)
1
1
(
x
,
y
)
2
2
D
Eliminate the square root by squaring both sides.
100 = (
x
 2) + (5  (3))
2
2
Simplify and solve for x.
100 = (
x
 2) + (5  (3))
2
2
100 =
x
 4
x
+ 4
+ (5 + 3)
2
2
100 =
x
 4
x
+ 4 +
64
2
100 =
x
 4
x
+
68
x
 4
x
+ 68
 100
= 0
2
2
x
 4
x
 32
= 0
2
Expand (x  2) .
2
Simplify (5(3)) .
2
Since 4 + 64 = 68,
Subtract both sides by 100 and switch sides.
Since 68100=32,
Factor.
(
x
 8)(
x
+ 4) = 0
Equate both factors to 0 and solve for x
x
 8 = 0 →
x
= 8
x
+ 4 = 0 →
x
= 4
Distance is always positive, so x = 8.
M
x + x
2
1
2
y + y
2
1
2
,
M
5
+
5
2
4
+ (
 2
)
2
,
M
0
2
2
2
,
M
(0,1)
Use the midpoint formula.
Identify (x , y ) and (x , y ) and plug these values into the formula.
1
1
2
2
(
x
,
y
)
1
1
(
x
,
y
)
2
2
M
x + x
2
1
2
y + y
2
1
2
,
Identify the given values.
Plug the given values and solve for x and y.
(
x
,
y
)
1
1
(
x
,
y
)
2
2
Midpoint
(6, 5)
x + x
2
1
2
y + y
2
1
2
6 =
5 =
x
+
4
2
y
+
(
7
)
2
6 =
5 =
12
=
x +
4
10
=
y
 7
x
= 8
y
=  3
2
2
2
2
Plane
NOTE that is different from .
DC
endpoint
D
C
endpoint
Full transcriptMidpoint Formula
Segment Addition Postulate
Defined Terms
The Three Undefined Terms
Segment Length and Midpoints
POint
Line
Line Segment
Ray
Other Terms
they served as the foundation of geometry that are used to define other terms
the most basic terms in Geometry that don’t require formal definitions
has no size or dimension
has location
M
represented by a dot
named using a capital letter
is a collection of points that extends infinitely in both directions
has no thickness
named using a small letter or by any two points on the line
n
X
Y
line
n
XY
YX
has two dimensions, the length and the width
has no height
flat surface that extends infinitely in both directions
O
A
C
B
named using a letter or by 3 points on the plane
plane
O
plane
ABC
point
M
terms with formal definition
part of a line
A
B
consists of two endpoints and all the points in between them
named using the two endpoints
segment
AB
segment
BA
AB
BA
C
D
part of a line that begins with an endpoint and ends infinitely
named using two points, the first point must be the endpoint
ray
CD
CD
Coplanar Points
Collinear Points
Parallel Lines
Postulate
points that are on the same plane
points that are on the same line
lines on the same plane that will not intersect
statement that is considered true without proof
X
Z
Y
XY
YZ
XZ
states that if we are given a line segment with endpoints
XY
X
and
Z
,
there is a point Y lying between points
X
and
Z
if and only if
+ YZ
= XZ
1. Use the segment addition postulate to answer this problem.
A
B
C
3
x
 6
6
x
+ 4
Given that AC=79,
find AB and BC.
STEP 1.
Find x using the segment addition postulate.
AB + BC = AC
Substitute A=3x6, B=6x+4 C=79.
(
3
x
 6
) + (
6
x
+ 4
) =
79
Add like terms.
9
x
 2 = 79
Add both sides by 2.
9
x
 2
+ 2
= 79
+ 2
9
x
= 81
Divide both sides by 9.
9
9
x
= 9
STEP 2.
Use
x
= 9 to find
AB
.
AB
= 3
x
 6
= 3(
9
)  6
= 27  6
AB
= 21
Substitute x=9.
Simplify.
STEP 3.
Use
x
= 9 to find
BC
.
BC
= 6
x
+ 4
= 6(
9
) + 4
= 54 + 4
BC
= 58
Substitute x=9.
Simplify.
STEP 3.
(Optional) Check the answers.
Use the segment addition postulate.
AB
+
BC
=
AC
21
+
58
=
79
79 = 79
Substitute the given AC=79 and our answers B=21 and BC=58. Simplify.
The statement is TRUE, then ours answers are correct!
D
= √(
x  x
) + (
y  y
)
2
1
2
2
1
2
The distance between two points
P
(
x
,
y
) and
P
(
x
,
y
) is given below
1
1
1
2
2
2
2. Determine the length of the line segment AB.
0
1
3
5
2
y
x
A
(1,3)
B
(5,2)
AB
= √(
x

x
) + (
y

y
)
1
2
1
2
2
2
= √(
5

1
) + (
2

3
)
= √4 + (5)
= √16 + 25
= √41
M
x + x
2
1
2
y + y
2
1
2
,
The midpoint
M
of a line segment with endpoint
A
(
x
,
y
) and
B
(
x
,
y
) can be found using the formula below
1
2
1
2
A
B
M
x
y
4. Find the midpoint of the two points (5, 4) and (5, 2).
5. Find x and y if (6, 5) is the midpoint of points (x, y) and (4, 7).
3. Find
x
so that the distance between the points (2, 3) and (
x
,5) is equal to 10.
2
2
2
2
Use the distance formula.
Identify (x , y ) and (x , y ). Plug the values in the formula.
1
1
2
2
(
x
,
y
)
1
1
(
x
,
y
)
2
2
Simplify.
Thus, the length of line segment AB is √41.
D
= √(
x

x
) + (
y

y
)
1
2
1
2
2
2
10
= √(
x

2
) + (
5
 (
3
))
2
2
Use the distance formula.
Identify (x , y ), (x , y ) and D. Plug the values in the formula.
1
1
2
2
(
x
,
y
)
1
1
(
x
,
y
)
2
2
D
Eliminate the square root by squaring both sides.
100 = (
x
 2) + (5  (3))
2
2
Simplify and solve for x.
100 = (
x
 2) + (5  (3))
2
2
100 =
x
 4
x
+ 4
+ (5 + 3)
2
2
100 =
x
 4
x
+ 4 +
64
2
100 =
x
 4
x
+
68
x
 4
x
+ 68
 100
= 0
2
2
x
 4
x
 32
= 0
2
Expand (x  2) .
2
Simplify (5(3)) .
2
Since 4 + 64 = 68,
Subtract both sides by 100 and switch sides.
Since 68100=32,
Factor.
(
x
 8)(
x
+ 4) = 0
Equate both factors to 0 and solve for x
x
 8 = 0 →
x
= 8
x
+ 4 = 0 →
x
= 4
Distance is always positive, so x = 8.
M
x + x
2
1
2
y + y
2
1
2
,
M
5
+
5
2
4
+ (
 2
)
2
,
M
0
2
2
2
,
M
(0,1)
Use the midpoint formula.
Identify (x , y ) and (x , y ) and plug these values into the formula.
1
1
2
2
(
x
,
y
)
1
1
(
x
,
y
)
2
2
M
x + x
2
1
2
y + y
2
1
2
,
Identify the given values.
Plug the given values and solve for x and y.
(
x
,
y
)
1
1
(
x
,
y
)
2
2
Midpoint
(6, 5)
x + x
2
1
2
y + y
2
1
2
6 =
5 =
x
+
4
2
y
+
(
7
)
2
6 =
5 =
12
=
x +
4
10
=
y
 7
x
= 8
y
=  3
2
2
2
2
Plane
NOTE that is different from .
DC
endpoint
D
C
endpoint