Lesson 4 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
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
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
6.
What would be the equation of the graph that is the set on all points that are
7.
Is the point
8.
If the graph is translated
Ready for More?
Is the equation
Takeaways
The equation of a circle with radius
The equation of a circle with radius
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.
1.
Factor.
a.
b.
2.
The arc is shown in green. Each indicated angle is the central angle that intercepts the given arc.
Given:
Find