# Lesson 4Circling TrianglesDevelop Understanding

## Learning Focus

Find the equation of a circle.

How do circles and right triangles relate?

## Open Up the Math: Launch, Explore, Discuss

Using the corner of a piece of colored paper and a ruler, cut a right triangle with a hypotenuse, like so:

Use this triangle as a pattern to cut three more just like it, so that you have a total of four congruent triangles.

### 1.

Choose one of the legs of the first triangle and label it , and label the other leg . What is the relationship between the three sides of the triangle?

### 2.

When you are told to do so, take your triangles up to the board, and place each of them on the coordinate axis like this:

Mark the point at the end of each hypotenuse with a pin.

### 3.

What shape is formed by the pins after the class has posted all of their triangles? Why would this construction create this shape?

### 4.

What are the coordinates of the pin that you placed in:

### 5.

Now that the triangles have been placed on the coordinate plane, some of your triangles have sides that are to the left of the origin, denoted as , or below the origin, denoted as . Is the relationship still true for these triangles? Why or why not?

### 6.

What would be the equation of the graph that is the set on all points that are away from the origin?

### 7.

Is the point on the graph? How about the point ? How can you tell?

### 8.

If the graph is translated to the right and up, what would be the equation of the new graph? Draw a diagram and explain how you found the equation.

## Takeaways

The equation of a circle with radius , centered at the origin:

The equation of a circle with radius and center :

## Lesson Summary

In this lesson, we derived the equation of a circle. We learned that the equation of a circle describes all the points a given distance from the center. Like the distance formula, it is based on the Pythagorean theorem.

## Retrieval

Factor.

### 2.

The arc is shown in green. Each indicated angle is the central angle that intercepts the given arc.

Given: and