Programmed tutorial: Find the Graph and Equation of a Circle through 3 Points: Example 1

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Try to solve this problem without using a graphing calculator.
Find the equation of the circle through the points
(-2, 2); (4, 2); (4, -6)

General Contents

Detailed Contents

Index

We’ll need the radius and the location of the center of the circle so we can use the equation of the general circle. We’ll be able to determine these from the triangle defined by these three points.

In order to see what type of triangle is determined by these three points, plot them on paper. Then check your diagram by clicking on “Next”.

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What type of triangle is it?

A right triangle.

What do we know about a circle through the vertices of a right triangle?

The hypotenuse is also the diameter.

So we need the length of the hypotenuse in order to get the radius.
How do we get the length of the hypotenuse?

We can use the definition of distance on the two endpoints of the hypotenuse.

Find this diameter, using (-2, 2) and (4, -6).

What can we conclude about the radius of the circle?

Since the radius is half of the diameter,
we have R = 5

How can we find the center of the circle?

Since (-2, 2) and (4, -6) are the ends of a diameter, the center must be halfway between them.

What method can we use to find that halfway point?

We can use the definition of the midpoint.

Apply that definition.

The coordinates of the midpoint are

Can we now find h and k, the coordinates of the center of the circle?

Yes.

What are they?

h = 1, k = -2

Use these results to write the equation of the circle.

becomes

Simplify this equation.

Add the circle to your graph. Then check your work by clicking “Next”.

alt="Java applet Circle and right triangle" Your browser is completely ignoring the <APPLET> tag!

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