PDF) Circles worksheet day #1 - Ms. Sullivan& #039;s CC Geometry ... - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

Part 1: Write an equation given Center and Radius

The standard formula for a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

1. Center $(2, -4)$, Radius $4$
* Plug in $h=2$ and $k=-4$: $(x - 2)^2 + (y - (-4))^2$ becomes $(x - 2)^2 + (y + 4)^2$.
* Square the radius: $4^2 = 16$.
* Equation: $(x - 2)^2 + (y + 4)^2 = 16$

2. Center $(-7, 1)$, Radius $15$
* Plug in $h=-7$ and $k=1$: $(x - (-7))^2 + (y - 1)^2$ becomes $(x + 7)^2 + (y - 1)^2$.
* Square the radius: $15^2 = 225$.
* Equation: $(x + 7)^2 + (y - 1)^2 = 225$

3. Center $(3, 0)$, Radius $1/3$
* Plug in $h=3$ and $k=0$: $(x - 3)^2 + (y - 0)^2$ becomes $(x - 3)^2 + y^2$.
* Square the radius: $(1/3)^2 = 1/9$.
* Equation: $(x - 3)^2 + y^2 = \frac{1}{9}$

4. Center $(-5, -3)$, Radius $3\sqrt{2}$
* Plug in $h=-5$ and $k=-3$: $(x - (-5))^2 + (y - (-3))^2$ becomes $(x + 5)^2 + (y + 3)^2$.
* Square the radius: $(3\sqrt{2})^2 = 3^2 \cdot (\sqrt{2})^2 = 9 \cdot 2 = 18$.
* Equation: $(x + 5)^2 + (y + 3)^2 = 18$

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Part 2: Write an equation given Center and a Point

First, find the radius using the distance formula between the center and the point: $r = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. Then square it to get $r^2$.

5. Center $(3, -4)$ passing through $(6, 2)$
* Find $r^2$: $(6 - 3)^2 + (2 - (-4))^2 = 3^2 + 6^2 = 9 + 36 = 45$.
* Equation uses center $(3, -4)$.
* Equation: $(x - 3)^2 + (y + 4)^2 = 45$

6. Center $(5, 1)$ passing through $(8, -2)$
* Find $r^2$: $(8 - 5)^2 + (-2 - 1)^2 = 3^2 + (-3)^2 = 9 + 9 = 18$.
* Equation uses center $(5, 1)$.
* Equation: $(x - 5)^2 + (y - 1)^2 = 18$

7. Center at origin $(0,0)$ passing through $(4, 3)$
* Find $r^2$: $(4 - 0)^2 + (3 - 0)^2 = 4^2 + 3^2 = 16 + 9 = 25$.
* Equation uses center $(0, 0)$.
* Equation: $x^2 + y^2 = 25$

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Part 3: Extension (Given endpoints of diameter)

To solve these, first find the Center (midpoint of the diameter) and then the Radius (distance from center to one endpoint).
* Midpoint Formula: $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

8. Endpoints $(5, 1)$ and $(3, -1)$
* Center: $(\frac{5+3}{2}, \frac{1+(-1)}{2}) = (\frac{8}{2}, \frac{0}{2}) = (4, 0)$.
* Radius squared ($r^2$): Distance from $(4,0)$ to $(5,1)$.
$(5-4)^2 + (1-0)^2 = 1^2 + 1^2 = 2$.
* Equation: $(x - 4)^2 + y^2 = 2$

9. Endpoints $(2, 1)$ and $(6, -3)$
* Center: $(\frac{2+6}{2}, \frac{1+(-3)}{2}) = (\frac{8}{2}, \frac{-2}{2}) = (4, -1)$.
* Radius squared ($r^2$): Distance from $(4,-1)$ to $(6,-3)$.
$(6-4)^2 + (-3 - (-1))^2 = 2^2 + (-2)^2 = 4 + 4 = 8$.
* Equation: $(x - 4)^2 + (y + 1)^2 = 8$

10. Endpoints $(4, -3)$ and $(2, 1)$
* Center: $(\frac{4+2}{2}, \frac{-3+1}{2}) = (\frac{6}{2}, \frac{-2}{2}) = (3, -1)$.
* Radius squared ($r^2$): Distance from $(3,-1)$ to $(4,-3)$.
$(4-3)^2 + (-3 - (-1))^2 = 1^2 + (-2)^2 = 1 + 4 = 5$.
* Equation: $(x - 3)^2 + (y + 1)^2 = 5$

Final Answer:
1. $(x - 2)^2 + (y + 4)^2 = 16$
2. $(x + 7)^2 + (y - 1)^2 = 225$
3. $(x - 3)^2 + y^2 = \frac{1}{9}$
4. $(x + 5)^2 + (y + 3)^2 = 18$
5. $(x - 3)^2 + (y + 4)^2 = 45$
6. $(x - 5)^2 + (y - 1)^2 = 18$
7. $x^2 + y^2 = 25$
8. $(x - 4)^2 + y^2 = 2$
9. $(x - 4)^2 + (y + 1)^2 = 8$
10. $(x - 3)^2 + (y + 1)^2 = 5$