Mth 111 – Coordinates, Midpoints, Distance, and Circles
ICA #2
Name: _______________
A. The midpoint of a line segment is the point halfway between the endpoints of the
segment.
1. Mark on the drawing the midpoint of each line segment below.
2. Find the coordinates of the endpoints and the midpoints for each of the following line
segments.
AB
(
Endpoints
,
)(
,
) (
Midpoint
,
)
CD
(
,
)(
,
) (
,
)
KL
(
,
)(
,
) (
,
)
MN
(
,
)(
,
) (
,
)
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 xcoordinates, x1 and x2, of the endpoints?
What is the mathematical expression for calculating
the y-coordinate of the midpoint using the ycoordinates of the endpoints?
Use your formulas to complete the table below.
Endpoints
Midpoint
TU
(
,
)(
,
) (
,
)
RS
(
,
)(
,
) (
,
)
4. Calculating the length of line segment AB is easy. But, how do we calculate the length
of the line segment TU????
Hint: On the graph, make a third point, W, at (-2, -3). What kind of triangle is TWU?
Recall the famous Pythagorean Theorem! It describes the relationship of the length of the
two sides, WT and WU, to the length of the hypotenuse, TU:
Formula? ________________________________
Use this formula to find the length of TU.
B. A circle is a collection of points that are all the equidistant from the center.
1. Plot the points (-1,2) and (3,-4) below and connect them with a line.
Think of this line as the diameter of a circle. Draw
the circle.
2. What is the center of the circle? (
,
) Length of the radius? _______
Justify mathematically whether the point (3,2) is on the circle. (Think about the distance
from the center to the point (3,2) - is it equal to the radius?)
3. Imagine a circle with center (h, k). Let (x,y) be any point on the circle. Find the length
of the radius in terms of h, k, x and y. (Hint: Use the Pythagorean theorem!)
4. For each formula of a circle what are the coordinates of the center and what is the
radius?
a)
Center is (
( x  3) 2  ( y  2) 2  7
,
) and radius = _____
b) ( x  4)2  ( y  5)2  16
Center is (
c) x 2  4 x  y 2  6 y  12
Center is (
,
) and radius = _____
,
) and radius = _____
d) x 2  8 x  y 2  13  0
Center is (
,
) and radius = _____
(Hint: Use the “complete the square” method.)
5. Write the formula of a circle with radius = 2 and centered at (1, -5).