Lesson 6 Circle Challenges Practice Understanding

Learning Focus

Apply understanding of circles and their equations to new situations.

How can we use algebra to find relationships within and between circles?

Open Up the Math: Launch, Explore, Discuss

Once Malik and Sapana started challenging each other with circle equations, they got a little more creative with their ideas. See if you can work out the challenges that they gave each other to solve. Be sure to support all of your answers by showing how you found them.

1.

Malik’s challenge: What is the equation of the circle with center $(- 13, - 16)$ that contains the point $(- 10, - 16)$ ?

2.

Sapana’s challenge: The points $(0, 5)$ and $(0, - 5)$ are the endpoints of the diameter of a circle. The point $(3, y)$ is on the circle. What are the possible values for $y$ ?

3.

Malik’s challenge: Find the equation of a circle with its center in the first quadrant and tangent to the lines $x = 8$ , $y = 3$ , and $x = 14$ .

4.

Sapana’s challenge: The points $(4, - 1)$ and $(- 6, 7)$ are the endpoints of the diameter of a circle. What is the equation of the circle?

5.

Malik’s challenge: Is the point $(5, 1)$ inside, outside, or on the circle $x^{2} - 6 x + y^{2} + 8 y = 24$ ? How do you know?

6.

Sapana’s challenge: The circle defined by ${(x - 1)}^{2} + {(y + 4)}^{2} = 16$ is translated $5$ units to the left and $2$ units down. Write the equation of the resulting circle.

7.

Malik’s challenge: There are two circles, the first with center $(3, 3)$ and radius $r_{1}$ , and the second with center $(3, 1)$ and radius $r_{2}$ .

Find two values of $r_{1}$ and $r_{2}$ so that the first circle is completely enclosed by the second circle.
Find one value of $r_{1}$ and one value of $r_{2}$ so that the two circles intersect at two points.
Find one value of $r_{1}$ and one value of $r_{2}$ so that the two circles intersect at exactly one point.

Ready for More?

In problem 7, you were asked to consider two circles, the first with center $(3, 3)$ and radius $r_{1}$ , and the second with center $(3, 1)$ and radius $r_{2}$ . Previously you found specific examples of $r_{1}$ and $r_{2}$ that meet the conditions of the statements. Now, your challenge is to find all the values, or the relationship between $r_{1}$ and $r_{2}$ , that meet the conditions of the statements.

What values of $r_{1}$ and $r_{2}$ make the first circle completely enclosed by the second circle?
Find the values of $r_{1}$ and $r_{2}$ so that the two circles intersect at exactly two points.
Find the values of $r_{1}$ and $r_{2}$ so that the two circles intersect at exactly one point.

Takeaways

Useful problem-solving strategies for circles:

Lesson Summary

In this lesson, we solved problems about circles that required us to use graphs and formulas such as the Pythagorean theorem, the distance formula, and the midpoint formula. We found it useful to use the equation of the circle to find points on the circle or to determine that a point is not on a circle. Sometimes it was useful to change forms of the equation to find more information about the circle from the equation.

Retrieval

1.

Find the perimeter of the polygon.

2.

Fill in the number that completes the square. Then write the trinomial as a square of a binomial. Leave the number that completes the square as a fraction.

a.

$x^{2} + 13 x + \underset{―}{}$

Factored form: $\underset{―}{}$

b.

$x^{2} + \frac{1}{3} x + \underset{―}{}$

Factored form: $\underset{―}{}$