Mth 111 – Coordinates, Midpoints, Distance
Section 3.1
Name: _______________
A. The midpoint of a line segment is the point halfway between the endpoints of the
segment.
1. Estimate the location of the midpoint of each line segment below with a dot.
2. Find the coordinates of the endpoints and the midpoints for each of the following line
segments.
AB
(
Endpoints
,
)(
,
CD
(
,
)(
,
) (
,
)
KL
(
,
)(
,
) (
,
)
MN
(
,
)(
,
) (
,
)
) (
Midpoint
,
)
Look for a pattern in the relationship
between the x-coordinates of the
endpoints and the x-coordinate of the
midpoint. Think of the midpoint as an
average of the endpoints.
3. After you see the pattern, write a mathematical expression for
calculating the x- coordinate of the midpoint using the x-coordinates,
x1 and x2, of the endpoints?
What is the mathematical expression for calculating the y-coordinate
of the midpoint using the y-coordinates of the endpoints?
Use your formulas to complete the table below.
Endpoints
Midpoint
Line TU
(
,
)(
,
)
(
,
)
Line RS
(
,
)(
,
) (
,
)
B. The length of a line segment can be thought of as the distance between two points.
1. What is the length of the line segment AB?
2. Much harder, what is the length of line segment TU?
a. On the graph mark a point called W at (-2, -3).
b. Draw line segments TW and UW.
c. What kind of triangle is TWU?
d. Let the length of WT be “a”, length WU be “b”, and length of TU be “c”.
3. Recall the famous Pythagorean Theorem! It describes the relationship of the
length of the two sides, a and b, to the length of the hypotenuse, c.
Formula? ________________________________
4. Use this formula to find the length of TU.
5. Use the formula again to find the length of RS
6. Suppose you have two points: (3, 8) and (-2, -5). Make a
rough graph and plot these points on the graph. What are
the coordinates of a third point that would create a right
triangle? _______ Draw the right triangle.
7. Suppose you have two points: (-2, 6) and (-7, 1). Make a
rough graph and plot these points on the graph. What are
the coordinates of a third point that would create a right
triangle? _______ Draw the right triangle.
8. Suppose you have two points: ( x 1, y 1) and ( x 2 , y 2 ) What
are the coordinates of a third point that would create a
right triangle? Hint: Make a graph.
9. Find the distance between each pair of points in questions 6, 7, and 8 (in
general). (The “distance between two points” is the same as the “length of the
line segments.”)