Lesson 4

Distances and Circles

  • Let’s build an equation for a circle.

Problem 1

Match each equation to its description.

A:

circle centered at \((0,\text-4)\) with a radius of 3

B:

circle centered at \((1,\text-4)\) with a radius of \(\sqrt{3}\)

C:

circle centered at \((1,4)\) with a radius of \(\sqrt{3}\)

D:

circle centered at \((1,0)\) with a radius of 3

E:

circle centered at \((1,\text-4)\) with a radius of 3

1:

\((x-1)^2+y^2=9\)

2:

\(x^2+(y+4)^2=9\)

3:

\((x-1)^2+(y-4)^2=3\)

4:

\((x-1)^2+(y+4)^2=9\)

5:

\((x-1)^2+(y+4)^2=3\)

Problem 2

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

A:

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

B:

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

C:

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

D:

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

Problem 3

  1. Plot the circles \(x^2+y^2=4\) and \(x^2+y^2=1\) on the same coordinate plane.
  2. Find the image of any point on \(x^2+y^2=4\) under the transformation \((x,y) \rightarrow \left(\frac{1}{2}x,\frac{1}{2}y\right)\).
  3. 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
\((1,1)\) \((\text-2,3)\)
\((6,1)\)  
\((3,5)\)  
(From Unit 6, Lesson 3.)

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?

A:

The image is congruent to the original triangle.

B:

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

C:

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

(From Unit 6, Lesson 3.)

Problem 6

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

A:

\((x,y)\rightarrow(3x,3y)\)

B:

\((x,y)\rightarrow(x-3,y-3)\)

C:

\((x,y)\rightarrow(x+3,y+3)\)

D:

\((x,y)\rightarrow(x-3,y)\)

E:

\((x,y)\rightarrow(x+3,y)\)

F:

\((x,y)\rightarrow(x,y-3)\)

G:

\((x,y)\rightarrow(x,y+3)\)

1:

Translate along the directed line segment from \((0,0)\) to \((\text-3,0)\).

2:

Translate along the directed line segment from \((0,0)\) to \((0,\text-3)\).

3:

Translate along the directed line segment from \((0,0)\) to \((3,0)\).

4:

Translate along the directed line segment from \((0,0)\) to \((0,3)\).

5:

Translate along the directed line segment from \((0,0)\) to \((3,3)\).

6:

Translate along the directed line segment from \((0,0)\) to \((\text-3,\text-3)\).

7:

Dilate using the origin as the center and a scale factor of 3.

(From Unit 6, Lesson 2.)

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?

A:

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

B:

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

C:

\(\frac19\)

D:

\(\frac13\)

(From Unit 5, Lesson 14.)