# Lesson 4

Distances and Circles

- Let’s build an equation for a circle.

### Problem 1

Match each equation to its description.

### Problem 2

Write an equation of a circle that is centered at \((\text-3,2)\) with a radius of 5.

\((x-3)^2+(y+2)^2=5\)

\((x+3)^2+(y-2)^2=5\)

\((x-3)^2+(y+2)^2=25\)

\((x+3)^2+(y-2)^2=25\)

### Problem 3

- Plot the circles \(x^2+y^2=4\) and \(x^2+y^2=1\) on the same coordinate plane.
- Find the image of any point on \(x^2+y^2=4\) under the transformation \((x,y) \rightarrow \left(\frac{1}{2}x,\frac{1}{2}y\right)\).
- What do you notice about \(x^2+y^2=4\) and \(x^2+y^2=1\)?

### Problem 4

\((x,y) \rightarrow (x-3,4-y)\) is an example of a transformation called a glide reflection. Complete the table using the rule.

Does this glide reflection produce a triangle congruent to the original?

input | output |
---|---|

\((1,1)\) | \((\text-2,3)\) |

\((6,1)\) | |

\((3,5)\) |

### Problem 5

The triangle whose vertices are \((1,1), (5,3),\) and \((4,5)\) is transformed by the rule \((x,y) \rightarrow (3x,3y)\). Is the image similar or congruent to the original figure?

The image is congruent to the original triangle.

The image is similar but not congruent to the original triangle.

The image is neither similar nor congruent to the original triangle.

### Problem 6

Match each coordinate rule to a description of its resulting transformation.

### Problem 7

A cone-shaped container is oriented with its circular base on the top and its apex at the bottom. It has a radius of 18 inches and a height of 6 inches. The cone starts filling up with water. What fraction of the volume of the cone is filled when the water reaches a height of 2 inches?

\(\frac{1}{729}\)

\(\frac{1}{27}\)

\(\frac19\)

\(\frac13\)