5-5
The Midpoint and Distance Formulas
Objective
Apply the formula for midpoint.
Use the distance formula to find the distance
between two points.
Holt McDougal Algebra 1
5-5
The Midpoint and Distance Formulas
Vocabulary
midpoint
Holt McDougal Algebra 1
5-5
The Midpoint and Distance Formulas
In Lesson 5-4, you used the coordinates of points
to determine the slope of lines. You can also use
coordinates to determine the midpoint of a line
segment on the coordinate plane.
The midpoint of a line segment is the point that
divides the segment into two congruent
segments. Congruent segments are segments
that have the same length.
You can find the midpoint of a segment by using
the coordinates of its endpoints. Calculate the
average of the x-coordinates and the average of
the y-coordinates of the endpoints.
Holt McDougal Algebra 1
5-5
The Midpoint and Distance Formulas
Holt McDougal Algebra 1
5-5
The Midpoint and Distance Formulas
Additional Example 1: Finding the Coordinates of a
Midpoint
Find the coordinates of the midpoint of GH
with endpoints G(–4, 3) and H(6, –2).
Write the
formula.
G(–4, 3)
Substitute.
H(6, -2)
Simplify.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Check It Out! Example 1
Find the coordinates of the midpoint of EF
with endpoints E(–2, 3) and F(5, –3).
Write the
formula.
E(–2, 3)
Substitute.
F(5, –3)
Simplify.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Additional Example 2:
Finding the Coordinates of an Endpoint
P is the midpoint of NQ. N has coordinates
(–5, 4), and P has coordinates (–1, 3). Find the
coordinates of Q.
Step 1 Let the coordinates of P equal (x, y).
Step 2 Use the Midpoint Formula.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Additional Example 2 Continued
Step 3 Find the
x-coordinate.
Find the
y-coordinate.
Set the
coordinates equal.
Multiply both
sides by 2.
–2 = –5 + x
+5 +5
3=x
Holt McDougal Algebra 1
6=4+y
Simplify.
Isolate the
variables.
−4 −4
Simplify.
2=y
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The Midpoint and Distance Formulas
Additional Example 2 Continued
The coordinates of Q are (3, 2).
Check Graph points Q and N
and midpoint P.
N (–5, 4)
P(–1, 3)
Q (3, 2)
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Check It Out! Example 2
S is the midpoint of RT. R has coordinates
(–6, –1), and S has coordinates (–1, 1) .
Find the coordinates of T.
Step 1 Let the coordinates of T equal (x, y) .
Step 2 Use the Midpoint Formula.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Check It Out! Example 2 Continued
Step 3 Find the
x-coordinate.
Find the
y-coordinate.
Set the
coordinates equal.
Multiply both
sides by 2.
–2 = –6 + x
+6
+6
4=x
Holt McDougal Algebra 1
Simplify.
Isolate the
variables.
Simplify.
2 = –1 + y
+1
+1
3=y
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The Midpoint and Distance Formulas
Check It Out! Example 2 Continued
The coordinates of T are (4, 3)
Check Graph points R and
S and midpoint T.
T(4, 3)
S(–1, 1)
R(–6, –1)
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
You can also use coordinates to find the
distance between two points or the length of a
line segment. To find the length of segment PQ,
draw a horizontal segment from P and a vertical
segment from Q to form a right triangle.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Remember!
The Pythagorean Theorem states that if a right
triangle has legs of lengths a and b and a
hypotenuse of length c, then a2 + b2 = c2.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Additional Example 3: Finding Distance in the
Coordinate Plane
Use the Distance Formula to find the distance, to
the nearest hundredth, from A(–2, –2) to B(4, 3).
Distance Formula
Substitute (4, –2) for (x1, y1)
and (3, –2) for (x2, y2).
Subtract.
Simplify powers.
Add.
Find the square root to the
nearest hundredth.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Additional Example 3 Continued
Use the Distance Formula to find the distance, to
the nearest hundredth, from A(–2, –2) to B(4, 3).
6
5
A (–2, –2)
Holt McDougal Algebra 1
B (4, 3)
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The Midpoint and Distance Formulas
Check It Out! Example 3
Use the Distance Formula to find the distance, to
the nearest tenth, from R(3, 2) to S(–3, –1).
Distance Formula
Substitute (3, 2) for (x1, y1)
and (-3, -1) for (x2, y2).
Add.
Simplify powers.
Add.
Find the square root to the
nearest hundredth.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Check It Out! Example 3 Continued
Use the Distance Formula to find the distance, to
the nearest tenth, from R(3, 2) to S(–3, –1).
6
3
S(–3, –1)
Holt McDougal Algebra 1
R(3, 2)
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The Midpoint and Distance Formulas
Additional Example 4: Application
Each unit on the map
represents 100 meters. To
the nearest tenth of a meter,
how far is it from the roller
coaster to the Ferris wheel?
Substitute.
It is 7.211  100 or
721.1 meters from
Add.
the roller coaster to
Simplify powers. Ferris Wheel.
Find the square root to the nearest
tenth.
Holt McDougal Algebra 1
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The Midpoint and Distance Formulas
Check It Out! Example 4
Jacob takes a boat from Pahokee to Clewiston.
To the nearest tenth of a mile, how far does he
travel?
Substitute.
Square.
Simplify powers.
d  17.7 miles
Holt McDougal Algebra 1
Find the square root to the
nearest tenth.