Mth 111 – Coordinates, Midpoints, Distance, and Circles Name: ______

ICA #2

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.

Endpoints Midpoint
AB ( , ) ( , ) ( , )
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 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

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) b)

Center is ( , ) and radius = _____ Center is ( , ) and radius = _____

c) d)

Center is ( , ) and radius = _____ 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).