Geometry Section 1-3 1112

Section 1-3
Distance and Midpoints

Essential Questions
How do you ﬁnd the distance between two points?
How do you ﬁnd the midpoint of a segment?

Vocabulary
1. Pythagorean Theorem:
2. Distance:

Vocabulary
1. Pythagorean Theorem: a2
+ b2
= c2
, where a and
b are legs of a right triangle, and c is the
hypotenuse
2. Distance:

Vocabulary
+ b2
= c2
, where a and
hypotenuse
2. Distance: The length of the segment formed
between two points

Vocabulary
+ b2
= c2
, where a and
hypotenuse
2. Distance: The length of the segment formed
between two points
d = (x2
− x1
)2
+ (y2
− y1
)2
for (x1
, y1
) and (x2
, y2
)

Vocabulary
3. Midpoint:
4. Segment Bisector:

Vocabulary
3. Midpoint: The point on a segment that is halfway
between the endpoints

Vocabulary
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ for (x1
, y1
) and (x2
, y2
)

Vocabulary
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ for (x1
, y1
) and (x2
, y2
)
4. Segment Bisector: Any segment, line, or plane
that intersects another segment at its midpoint

Example 1
Use the number line to ﬁnd DJ.
−4 −3 −2 −1 0 1 2 3 4
J D

Example 1
−4 −3 −2 −1 0 1 2 3 4
J D
x2
− x1

Example 1
−4 −3 −2 −1 0 1 2 3 4
J D
x2
− x1
4 − (−4)

Example 1
−4 −3 −2 −1 0 1 2 3 4
J D
x2
− x1
4 − (−4) 4 + 4

Example 1
−4 −3 −2 −1 0 1 2 3 4
J D
x2
− x1
4 − (−4) 4 + 4 8

Example 1
−4 −3 −2 −1 0 1 2 3 4
J D
x2
− x1
4 − (−4) 4 + 4 8 8

Example 1
−4 −3 −2 −1 0 1 2 3 4
J D
x2
− x1
4 − (−4) 4 + 4 8 8
8 units

Example 2
Graph A(3, 2) and B(6, 8). Then use the
Pythagorean Theorem to ﬁnd AB.

Example 2
x
y

Example 2
x
y
A

Example 2
x
y
A
B

Example 2
x
y
A
B
3
6

Example 2
x
y
A
B
3
6
a2
+ b2
= c2

Example 2
x
y
A
B
3
6
a2
+ b2
= c2
32
+ 62
= c2

Example 2
x
y
A
B
3
6
a2
+ b2
= c2
32
+ 62
= c2
9 + 36 = c2

Example 2
x
y
A
B
3
6
a2
+ b2
= c2
32
+ 62
= c2
9 + 36 = c2
45 = c2

Example 2
x
y
A
B
3
6
a2
+ b2
= c2
32
+ 62
= c2
9 + 36 = c2
45 = c2
45 = c2

Example 2
x
y
A
B
3
6
a2
+ b2
= c2
32
+ 62
= c2
9 + 36 = c2
45 = c2
45 = c2
c ≈ 6.708203933 units

Example 3
Use the distance formula to ﬁnd the distance
between A(3, 2) and B(6, 8).

Example 3
d = (x2
− x1
)2
+ (y2
− y1
)2

Example 3
d = (x2
− x1
)2
+ (y2
− y1
)2
= (6 − 3)2
+ (8 − 2)2

Example 3
d = (x2
− x1
)2
+ (y2
− y1
)2
= (6 − 3)2
+ (8 − 2)2
= 32
+ 62

Example 3
d = (x2
− x1
)2
+ (y2
− y1
)2
= (6 − 3)2
+ (8 − 2)2
= 32
+ 62
= 9 + 36

Example 3
d = (x2
− x1
)2
+ (y2
− y1
)2
= (6 − 3)2
+ (8 − 2)2
= 32
+ 62
= 9 + 36
= 45

Example 3
d = (x2
− x1
)2
+ (y2
− y1
)2
= (6 − 3)2
+ (8 − 2)2
= 32
+ 62
= 9 + 36
= 45
≈ 6.708203933 units

Example 4
Find the midpoint of AB for points A(3, 2) and
B(6, 8).

Example 4
B(6, 8).
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟

Example 4
B(6, 8).
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ =
3 + 6
2
,
2 + 8
2
⎛
⎝
⎜
⎞
⎠
⎟

Example 4
B(6, 8).
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ =
3 + 6
2
,
2 + 8
2
⎛
⎝
⎜
⎞
⎠
⎟
=
9
2
,
10
2
⎛
⎝
⎜
⎞
⎠
⎟

Example 4
B(6, 8).
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ =
3 + 6
2
,
2 + 8
2
⎛
⎝
⎜
⎞
⎠
⎟
=
9
2
,
10
2
⎛
⎝
⎜
⎞
⎠
⎟ =
9
2
,5
⎛
⎝
⎜
⎞
⎠
⎟

Example 4
B(6, 8).
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ =
3 + 6
2
,
2 + 8
2
⎛
⎝
⎜
⎞
⎠
⎟
=
9
2
,
10
2
⎛
⎝
⎜
⎞
⎠
⎟ =
9
2
,5
⎛
⎝
⎜
⎞
⎠
⎟ or 4.5,5( )

Example 5
Find the coordinates of U if F(−2, 3) is the
midpoint of UO and O has coordinates of (8, 6).

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8
x = −12

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8
x = −12
3 =
y + 6
2

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8
x = −12
3 =
y + 6
2
i2

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8
x = −12
3 =
y + 6
2
i22i

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8
x = −12
3 =
y + 6
2
i22i
6 = y + 6

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8
x = −12
3 =
y + 6
2
i22i
6 = y + 6
y = 0

Example 5
M =
x1
+ x2
2
,
y1
+ y2
2
⎛
⎝
⎜
⎞
⎠
⎟ (−2,3) =
x + 8
2
,
y + 6
2
⎛
⎝
⎜
⎞
⎠
⎟
−2 =
x + 8
2
i2(2)i( )
−4 = x + 8
x = −12
3 =
y + 6
2
i22i
6 = y + 6
y = 0
U(−12,0)

Example 6
Find PQ if Q is the midpoint of PR.
2x + 3 4x − 1
P Q R

Example 6
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1

Example 6
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1
4 = 2x

Example 6
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2

Example 6
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3

Example 6
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
PQ = 2(2) + 3

Example 6
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
PQ = 2(2) + 3
PQ = 4 + 3

Example 6
2x + 3 4x − 1
P Q R
2x + 3 = 4x − 1
4 = 2x
x = 2
PQ = 2x + 3
PQ = 2(2) + 3
PQ = 4 + 3
PQ = 7 units

Problem Set
p. 30 #1-11, 19-31, 47, 49, 53 (odds only)
“Learn from yesterday, live for today, hope for
tomorrow. The important thing is not to stop
questioning.” - Albert Einstein

Geometry Section 1-3 1112

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Geometry Section 1-3 1112