Lesson 1 Circling Triangles Develop 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 $6 -inch$ 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 $x$ , and label the other leg $y$ . 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:

a.

The first quadrant?

b.

The second quadrant?

c.

The third quadrant?

d.

The fourth quadrant?

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 $- x$ , or below the origin, denoted as $- y$ . Is the relationship $x^{2} + y^{2} = 6^{2}$ 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 $6 inches$ away from the origin?

7.

Is the point $(0, - 6)$ on the graph? How about the point $(3, 5.193)$ ? How can you tell?

8.

If the graph is translated $3 units$ to the right and $2 units$ up, what would be the equation of the new graph? Draw a diagram and explain how you found the equation.

Ready for More?

Is the equation $x^{2} + y^{2} = 36$ a function? Explain your answer.

Takeaways

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

The equation of a circle with radius $r$ and center $(h, k)$ :

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

1.

Factor.

a.

$225 x^{2} - 16 y^{2}$

b.

$225 x^{2} - 120 x y + 16 y^{2}$

2.

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

Given: $r = 24 mm$ and $\overset{⌢}{A B} = 40 π mm$

Find $m ∠ A C B$ in radians.