Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Apply the formula for midpoint.Use the distance formula to find the distancebetween two points.
Objective
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
midpoint
Vocabulary
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
Find the coordinates of the midpoint of GH with endpoints G(–4, 3) and H(6, –2).
Substitute.
Write the formula.
Simplify.
Additional Example 1: Finding the Coordinates of a Midpoint
G(–4, 3)
H(6, -2)
Holt McDougal Algebra 1
5-5 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.
Substitute.
Simplify.
E(–2, 3)
F(5, –3)
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Additional Example 2: Finding the Coordinates of an Endpoint
Step 1 Let the coordinates of P equal (x, y).
Step 2 Use the Midpoint Formula.
P is the midpoint of NQ. N has coordinates (–5, 4), and P has coordinates (–1, 3). Find the coordinates of Q.
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Additional Example 2 Continued
Multiply both sides by 2.
Isolate the variables.
–2 = –5 + x+5 +5
3 = x
6 = 4 + y
−4 −4
Simplify. 2 = y
Set the coordinates equal.
Step 3 Find the x-coordinate.
Find the y-coordinate.
Simplify.
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
The coordinates of Q are (3, 2).
N (–5, 4)P(–1, 3)
Q (3, 2)
Check Graph points Q and N and midpoint P.
Additional Example 2 Continued
Holt McDougal Algebra 1
5-5 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
5-5 The Midpoint and Distance Formulas
Check It Out! Example 2 Continued
Multiply both sides by 2.
Simplify.
Simplify.
Isolate the variables.
–2 = –6 + x
+6 +6
2 = –1 + y
+1 +1
4 = x 3 = y
Set the coordinates equal.
Step 3 Find the x-coordinate.
Find the y-coordinate.
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
The coordinates of T are (4, 3)
R(–6, –1)
T(4, 3)
S(–1, 1)
Check Graph points R and S and midpoint T.
Check It Out! Example 2 Continued
Holt McDougal Algebra 1
5-5 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
5-5 The Midpoint and Distance Formulas
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.
Remember!
Holt McDougal Algebra 1
5-5 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
5-5 The Midpoint and Distance Formulas
Additional Example 3 Continued
A (–2, –2)
B (4, 3)6
5
Use the Distance Formula to find the distance, to the nearest hundredth, from A(–2, –2) to B(4, 3).
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Check It Out! Example 3Use 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
5-5 The Midpoint and Distance Formulas
Check It Out! Example 3 Continued
S(–3, –1)
R(3, 2)63
Use the Distance Formula to find the distance, to the nearest tenth, from R(3, 2) to S(–3, –1).
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
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?
Additional Example 4: Application
Substitute.
Add.
Simplify powers.
Find the square root to the nearest tenth.
It is 7.211 100 or 721.1 meters from the roller coaster to Ferris Wheel.
Holt McDougal Algebra 1
5-5 The Midpoint and Distance Formulas
Check It Out! Example 4Jacob takes a boat from Pahokee to Clewiston. To the nearest tenth of a mile, how far does he travel?
d 17.7 miles
Substitute.
Square.
Simplify powers.
Find the square root to the nearest tenth.
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