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Dilation

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Recognizing Dilations

Have you ever thought about a log cabin? Take a look at this dilemma about building log cabins?

Sherri decided to do her project on a log cabin. Mrs. Patterson suggested that she choose a time period to work with because log cabins have been being built for a long time. Years ago, they were quite small, but today, log cabins can be designer homes too.

“Mrs. Patterson, I am going to focus on a log cabin from the 1800’s,” Sherri said taking out a book that she found at the library on log cabins.

“That is a great idea. What was the average size of a log cabin in 1800?” Mrs. Patterson asked.

“What do you mean?”

“I mean the square footage. How many feet wide was the average house and how many feet long?” Mrs. Patterson explained.

“Oh, I get it. The average house was 20 \times 40 \ feet . So the average was 800 square feet,” Sherri said.

“Terrific, now make sure that your plan shows that,” Mrs. Patterson said walking away.

Sherri is puzzled. She knows that the shape of the log cabin is a rectangle given the length and width. She knows the area of the house. To create a plan, she will need to create a dilation. Sherri decides that she will use a scale factor of \frac{1}{16} . Given this information, what will the dimensions be of her house plan?

Use this Concept to learn about dilations and figure out the dimensions of the house by the end of it.

Guidance

There are all kinds of transformations. We can flip or reflect a figure, translate or slide a figure and rotate a figure. We can also stretch or shrink a figure to create a new one. This is called a dilation .

A dilation is a transformation created by a scale factor.

We can create a dilation that is smaller or larger than the original figure. Either way, a similar figure is created through a dilation.

Let’s think about scale factors for a minute.

The scale factor is the ratio that determines the proportional relationship between the sides of similar figures.

For the pairs of sides to be proportional to each other, they must have the same scale factor. In other words, similar figures have congruent angles and sides with the same scale factor. A scale factor of two means that each side of the larger figure is exactly twice as long as the corresponding side in the smaller figure.

When we compare the corresponding sides of a figure, we can figure out the scale factor of that figure.

A figure has a side length of 3 feet. What would be the corresponding side length of the next figure is the scale factor was 4?

Let’s think about this. We know the length of one of the sides of the first figure and we know the scale factor. To figure out the new length, we can multiply the scale factor times the first length.

3 \times 4 = 12

The length of the corresponding side of the second figure is 12 feet.

When we have a figure that is larger than the original, we have a scale factor that is greater than one. If we have a figure that is smaller than the original, then we have a scale factor that is less than one or a fraction.

A figure has a side length of 5 meters. What would be the corresponding side length of the new figure if the scale factor is \frac{1}{2} ?

To figure this out, we have to take the given length of the first figure and divide it in half. This will give us the corresponding length of the second figure.

5 \left(\frac{1}{2}\right) = 2.5

The length of the corresponding side will be 2.5 meters.

Now that you understand dilations, we can look at how to work with them on the coordinate plane. Once again, we will be using coordinate notation to describe the different dilations that are created on the coordinate plane.

Let’s look at this figure and then see how we can graph the dilation of it.

Graphing dilations of geometric figures is actually fairly easy to do when we know the scale factor. We simply multiply both coordinates for each vertex by the scale factor to produce new coordinates.

Suppose we want to make an enlargement of the rectangle above using a scale factor of 3. We need to multiply each coordinate by 3.

& (-2,-3) \ (-2,3) \ (2,3) \ (2,-3)\\& \times 3\\& (-6,-9) \ (-6,9) \ (6,9) \ (6,-9)

Now we can graph it on the coordinate plane.

We can create a reduction too. We create a reduction by dividing each coordinate by the scale factor. This will give us the new measurements of the figure.

Find each new measurement given the scale factor.

A quadrilateral with side measures of 6, 15, 27, 30.

Example A

A scale factor of \frac{1}{3} .

Solution: 2, 5, 9, 10

Example B

A scale factor of \frac{1}{2} .

Solution: 3, 7.5, 13.5, 15

Example C

A scale factor of 2.

Solution: 12, 30, 54, 60

Now let's go back to the dilemma from the beginning of the Concept.

To solve this problem, we begin with the actual dimensions of the log cabin. The log cabin has real – world dimensions of 20 \times 40 \ feet .

Sherri is using a scale factor of \frac{1}{16} . That means that the dilation will be a reduction. We divide both dimensions by 16.

20 \div 16 &= 1.25 \ ft.\\40 \div 16 &= 2.5 \ ft

The dimensions of Sherri’s plan will be 1.25 \ ft \ wide \times 2.5 \ ft \ long .

Vocabulary

Dilation
to reduce or enlarge a figure according to a scale factor.
Scale Factor
the ratio that compares the lengths of corresponding sides to each other. That comparison is the scale factor.

Guided Practice

Here is one for you to try on your own.

Graph a reduction of the following figure if the scale factor is \frac{1}{2} .

Solution

Notice that each of the original coordinates were divided by two to create the coordinates of the reduction.

(2, 4) \div 2 &= (1, 2)\\(8, -4) \div 2 &= (4, -2)\\(-6, -2) \div 2 &= (-3, -1)

Video Review

Practice

Directions: Use each scale factor to determine the new dimensions of each figure.

  1. A triangle with side measures of 4, 5, 9 and a scale factor of 2.
  2. A triangle with side measures of 4, 5, 9 and a scale factor of 3.
  3. A triangle with side measures of 4, 5, 9 and a scale factor of 4.
  4. A triangle with side measures of 8, 10, 14 and a scale factor of 2.
  5. A triangle with side measures of 8, 10, 14 and a scale factor of 4.
  6. A triangle with side measures of 2, 4, 6 and a scale factor of 2.
  7. A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of \frac{1}{2}
  8. A quadrilateral with side measures of 12, 16, 20, 24 and a scale factor of \frac{1}{4}
  9. A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of 2
  10. A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of 3
  11. A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of 4
  12. A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of \frac{1}{3}
  13. A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of 2
  14. A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of 3
  15. A quadrilateral with side measures of 8, 12, 16, 24 and a scale factor of \frac{1}{4}
  16. A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of \frac{1}{2}

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