Lesson 3

Types of Transformations

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Problem 1

Complete the table and determine the rule for this transformation.

input output
\((2,\text-3)\) \((\text-3,2)\)
\((4,5)\) \((5,4)\)
\((\rule{.5cm}{0.4pt},4)\) \((4,0)\)
\((1,6)\)  
  \((\text-1,\text-2)\)
\((x,y)\)  

Solution

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Problem 2

Write a rule that describes this transformation.

original figure image
\((5,1)\) \((2,\text-1)\)
\((\text-3,4)\) \((\text-6,\text-4)\)
\((1,\text-2)\) \((\text-2,2)\)
\((\text-1,\text-4)\) \((\text-4,4)\)

Solution

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Problem 3

Select all the transformations that produce congruent images.

A:

dilation

B:

horizontal stretch

C:

reflection

D:

rotation

E:

translation

Solution

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Problem 4

Here are some transformation rules. For each transformation, first predict what the image of triangle \(ABC\) will look like. Then compute the coordinates of the image and draw it.

  1. \((x,y) \rightarrow (x-4,y-1)\)
  2. \((x,y) \rightarrow (y,x)\)
  3. \((x,y) \rightarrow (1.5x,1.5y)\)
Triangle ABC. A at 2 comma 3, B at 4 comma 2, C at 3 comma 5.

Solution

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(From Unit 6, Lesson 2.)

Problem 5

A cylinder has radius 3 inches and height 5 inches. A cone has the same radius and height.

  1. Find the volume of the cylinder.
  2. Find the volume of the cone.
  3. What fraction of the cylinder’s volume is the cone’s volume?

Solution

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(From Unit 5, Lesson 13.)

Problem 6

Reflect triangle \(ABC\) over the line \(x=\text-2\). Call this new triangle \(A’B’C’\). Then reflect triangle \(A’B’C’\) over the line \(x=0\). Call the resulting triangle \(A''B''C''\).

Describe a single transformation that takes \(ABC\) to \(A''B''C''\).

Triangle ABC graphed on coordinate plane. A at 1 comma 1, B at 2 comma -1, C at 3 comma 0.

Solution

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(From Unit 6, Lesson 1.)

Problem 7

In the construction, \(A\) is the center of one circle, and \(B\) is the center of the other. 

Two circles, centered at A and B, each pass through the center of the other and intersect at C and D. Line AB extends horizontally across both circles. Radii AC, BC, AD and BD form rhombus ACBD.

Explain why segments \(AC\), \(BC\), \(AD\), \(BD\), and \(AB\) have the same length.

Solution

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(From Unit 1, Lesson 2.)