Section 6.7

Integration: Geometry

Midpoint of a Line Segment

What you should learn

To find the coordinates of the midpoint of a line segment in the coordinate plane

NCTM Curriculm Standards 2 - 4, 6 - 10

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Midpoint

Introduction: Maria Hernandez made a frame for one of her watercolor paintings in a class offered at the community college. Her instructor suggested that she stabilize her frame by adding crossbars connecting the consecutive sides. One of the boooks she used as a reference showed the crossbars inserted so that the bars are placed on the sides of the frame at the midpoint of each side. The midpoint of a line segment is the point that is halfway between the endpoints of the segment.

Maria decided to use some grid paper to model her frame and the crossbars she is going to insert. Each square of the paper represented 2 inches of her frame. To find the midpoint of the interior vertical edge, she counted how many units it was from one corner to the next and divided by 2. The inside of the frame is 16 inches high, so the midpoint is 8 inches form the corner. She counted 4 units on the grid and placed a bullet on her drawing. She used a similar method to find the midpoint of the interior horizontal edge. The inside is 20 inches wide, so the midpoint is 10 inches from the end.

Suppose we place Maria's frame drawing on a coordinate plane. Let's list the coordinates of the corners of the frame and the midpoints of the sides. What patter do you notice?

A (2, 2)

B (22, 2)

C (22, 18)

D (2, 18)

Midpoint of AB: (12, 2)

Midpoint of BC: (22, 10)

Midpoint of CD: (12, 18)

Midpoint of AD: (2, 10)

Notice that the coordinates of the midpoint of AB are the mean of the corresponding coordinates of A and B.

The woodworking examples uses the midpoints of horizontal and vertical line segments. The activity below applies to any segment. (Go to activity now)...

This activity suggests the following rule for finding the midpoint of any segment, given the coordinates of its endpoints.

Midpoint of a Line Segment on a Coordinate Plane: The coordinates of the midpoint of a line segment whose endpoints are at (x1, y1) and (x2, y2) are given by ([x1 + x2]/2, [y1 + y2]/2).

Exercise 1: If the vertices of parallelogram WXYZ are W (3, 0), X (9, 3), Y (7, 10), and Z (1, 7), prove that the diagonals bisect each other. That is, prove they intersect at their midpoints. (Use your problem solve skills and graph the parallelogram).

If you know the midpoint and one endpoint of a segment, you can find the other endpoint.

Exercise 2: The center of a circle is M (0, 2), and the endpoint of one of its radii is A (-6, -4). If AB is a diameter of the circle, what are the coordinates of B?

Use the midpoint formula and substitude the values you know...

Activity: Midpoint of a Line Segment

Materials: Grid paper, ruler, and colored pencils

Create a coordinate plane on your grid paper.

Use a ruler to draw any line segment on the coordinate plane. The segment should not be vertical or horizontal. Label the endpoints of the segment A and B with their coordinates.

Hold the paper up to the light and fold it so that A and B coincide. Crease the paper at the fold.

Unfold the paper. Label the point M at which the crease meets the segment and write the coordinates of M.

Use a different color pencil to draw a vertical line through the lower endpoint of your segment. Then draw a horizontal line through the upper endpoint of your segment. Label the point of intersection P.

Write the coordinates of the midpoints of AP and BP

How do the x-coordinate of the vertical segment and the y-coordinate of the horizontal segment compare with the coordinates of M?

YOUR TURN

a. Creater another coordinate plane and draw a segment with a different slope from the first one you drew.

b. Repeat the activity with this segment. What are your results?

c. Write a general rule for finding the midpoint of any segment.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 13 - 39 odd, 40, 41, 43 - 51

Alternative Homework: Enriched: 14 - 38 even, 40 - 51

Extra Practice: Students book page 771 Lesson 6-7

Extra Practice Worksheet: Click Here.