# 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)/28  =  2 + x6  =  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. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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