THE MIDPOINT AND DISTANCE FORMULAS
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.
Midpoint Formula
Let A(x1, y1) and B(x2, y2).
The midpoint M of the line segment AB is
Finding the Coordinates of a Midpoint
Example 1 :
Find the coordinates of the midpoint of the line segment CD with endpoints C(-2, -1) and D (4, 2).
Solution :
Write the formula.
= M[(x1 + x2)/2, (y1 + y2)/2]
Substitute (-2, -1) for (x1, y1) and (4, 2) for (x2, y2).
= M[(-2 + 4)/2, (-1 + 2)/2]
= M(2/2, 1/2)
= M(1, 1/2)
Finding the Coordinates of an Endpoint
Example 2 :
M is the midpoint of the line segment AB. A has coordinates (2, 2), and M has coordinates (4, -3). Find the coordinates of B.
Solution :
Step 1 :
Let the coordinates of B equal (x, y).
Step 2 :
Use the Midpoint Formula.
(4, -3) = [(2 + x)/2, (2 + y)/2]
Step 2 :
|
Find the x-coordinate. 4 = (2 + x)/2 8 = 2 + x 6 = x |
Find the y-coordinate. -3 = (2 + y)/2 -6 = 2 + y -8 = y |
The coordinates of B are (6, –8).
Check :
Graph points A and B and midpoint M.
Point M appears to be the midpoint of the line segment AB.
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 as shown below.
Pythagorean Theorem :
c2 = a2 + b2
Solve for c. Use the positive square root to represent distance.
c = √(a2 + b2)
This equation represents the Distance Formula.
Distance Formula
In a coordinate plane, formula to find the distance between two points (x1, y1) and (x2, y2) is
Finding Distance in the Coordinate Plane
Example 3 :
Use the Distance Formula to find the distance, to the nearest hundredth, from A(-2, 3) to B(2, -2).
Solution :
Distance Formula :
d = √[(x2 - x1)2 + (y2 - y1)2]
Substitute (-2, 3) for (x1, y1) and (2, -2) for (x2, y2).
d = √[(2 + 2)2 + (-2 - 3)2]
Simplify.
d = √[42 + (-5)2]
d = √[16 + 25]
d = √41
d ≈ 6.40
The distance between from A(-2, 3) to B(2, -2) is about 6.40 units.
Geography Application
Example 4 :
Each unit on the map of Lake Okeechobee represents 1 mile. Kemka and her father plan to travel from point A near the town of Okeechobee to point B at Pahokee. To the nearest tenth of a mile, how far do Kemka and her father plan to travel?
Solution :
Distance Formula :
d = √[(x2 - x1)2 + (y2 - y1)2]
Substitute (33, 13) for (x1, y1) and (22, 39) for (x2, y2).
d = √[(33 - 22)2 + (13 - 39)2]
Simplify.
d = √[112 + (-26)2]
d = √[121 + 676]
d = √797
d ≈ 28.2
Kemka and her father plan to travel about 28.2 miles.
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