Minds On

Stretch your perspective

Imagine you are creating a poster to promote an upcoming school basketball game. There are lots of great images to choose from and you add your selections to the poster. Unfortunately, when you try to change the size of the image of a basketball, it becomes distorted.

The following are three images of the same basketball.

What's wrong with these images?

There is the original image.

A regular basketball ball

The other two images are distorted.

  • One image has been pulled too much horizontally.
  • The other has been pulled too much vertically.
Two distorted basketballs, one pulled too much vertically and another horizontally. Radius is not same.

Making connections

Have you ever had one or more images distorted before in your own life?

Open a blank document on a computer and insert a picture or clip art image.

Practice pulling the image from different sides. Record what you know about what happened to this image. How can you make the image larger or smaller without causing it to be distorted?

Press ‘Remember’ to reveal information about dilation.

Remember:

  • In a dilation, the orientation, proportions, and shape remain the same after it has been transformed (enlarged or reduced by a certain scale factor).
  • A dilated image is similar to the original.
  • Similar figures have the same shape (angles are congruent) and their corresponding sides are proportional.
  • If the width of a dilated rectangle is now twice as long, so too is its length. (Scale factor of 2)

Action

How to perform a dilation

Activity one: dilations with a scale factor of 2

We will show dilations on a grid so that side lengths can easily be measured. Remember that scale factor refers to how many times larger (or smaller) we are making lengths for an image.

  • A rectangle with vertices labeled ABCD has side lengths of 5 units long by 3 units wide.
  • It is dilated by a scale factor of 2. It means that the side lengths and width are twice that of the original rectangle.
  • Objects can be dilated from a point outside the shapes or from one of the vertices of the shape. This point is called the centre of dilation.

The following two diagrams show these two dilation methods.

A rectangle with vertices labeled ABCD has side lengths of 5 units long by 3 units wide. It is dilated by a scale factor of 2 through a centre of dilation that is 4 units up and 4 units 3 the right. 4 lines are drawn diagonally from the centre of dilatation to the original rectangle to form the resulting rectangle. The resulting rectangle A prime B prime C prime D prime is 10 units long and 6 units wide. This rectangle is 4 units down and 4 units right of the original rectangle.

A rectangle with vertices labeled ABCD has side lengths of 5 units long by 3 units wide. It is dilated by a scale factor of 2 through a centre of dilation that is located at vertex A. The resulting rectangle A prime B prime C prime D prime is 10 units long and 6 units wide.

For both diagrams:

  • What do you notice about the side lengths of the rectangle when comparing the original with the one that is dilated?
  • What are the new side lengths?
  • How might you describe what happens when you perform a dilation by a scale factor of 2?

Record your thinking orally, digitally or in print.

Activity two: dilations with a scale factor of 3

In this second example, there is a right triangle with height of 4 units and a base of 3 units. Use a scale factor of 3 to perform a dilation. You may dilate from a point outside the image or use any corner of the original triangle.

  • What are the new side lengths?
  • What do you notice about the side lengths of the original triangle and the new triangle?
  • The side that's on a slant (the hypotenuse) isn't easy to measure using the grid. If you were told the slant (hypotenuse) on the original is 5 units long, how long do you think it might be on the dilated image?
  • How might you describe what happens when you perform a dilation with a scale factor of 3?
A right triangle with a height of 4 units and a base of 3 units. A line is drawn joining the two sides.

Press ‘Hint’ to reveal a suggested action.

Measure the diagonals of both figures and compare their lengths.

Activity Three: Dilations with Other Shapes

Using a virtual manipulative or graph paper, plot a trapezoid like the one below or record a detailed description (audio or written).

  • Perform a dilation of this trapezoid with a scale factor of 2.
  • What do you notice about the original shape and the new shape?

A trapezoid with bottom 6 units, a height is drawn straight up on the left side 4 units long. The top is 2 units long extending horizontally from the top of the left side. The last side goes on a slant from the right side of the top to the right side of the bottom. This is drawn on a grid.

  • Re-create the original trapezoid using a virtual manipulative or graph paper or record a detailed description of what is happening. Then, perform a dilation of the trapezoid with a scale factor of 1 2  . What do you notice about the original shape and the new shape?
  • How might you describe the relationship between the original trapezoid and the dilated trapezoid?

Consolidation

Reflecting on dilations

Create a 4-sided figure. Then, use a scale factor of 3 to dilate the figure. Be sure to use different symbols to identify the vertices of the new shape.

Record your work digitally or in print, or record a detailed description of your figure, the transformation, and the new shape. Use the following questions to guide your recording.

  • What do you notice about the original shape compared to the image?
  • Can you use your creation as a work of art? Explain your thinking.
  • Can you think of some professions or scenarios where the applications of dilations are used?
  • How do you know your dilations have been performed correctly?

Press ‘Hint’ to check your answer to the last question.

How do you know your dilations have been performed correctly? Measure the lengths across the diagonals of the original shape and the dilated image.

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

I feel...

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

Connect with a TVO Mathify tutor

Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.

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