Lesson 4

Distances and Circles

Problem 1

Match each equation to its description.

Solution

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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:

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

Solution

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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 \(x^2+y^2=1\)?

Solution

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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
\((1,1)\) \((\text-2,3)\)
\((6,1)\)  
\((3,5)\)  

Solution

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(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.

Solution

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(From Unit 6, Lesson 3.)

Problem 6

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

Solution

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(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:

\(\frac13\)

Solution

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(From Unit 5, Lesson 14.)