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Lesson 3Dilations with no Grid

Let’s dilate figures not on grids.

Learning Targets:

  • I can apply a dilation to a polygon using a ruler.

3.1 Points on a Ray

  1. Find and label a point C on the ray whose distance from A is twice the distance from B to A .
  2. Find and label a point D on the ray whose distance from A is half the distance from B to A .
Ray AB is shown

3.2 Dilation Obstacle Course

open applet in presentation mode

  1. Dilate B using a scale factor of 5 and A as the center of dilation. Which point is its image?

  2. Using H as the center of dilation, dilate G so that its image is E . What scale factor did you use?

  3. Using H as the center of dilation, dilate E so that its image is G . What scale factor did you use?

  4. To dilate F so that its image is B , what point on the diagram can you use as a center?

  5. Dilate H using A as the center and a scale factor of \frac{1}{3} . Which point is its image?

  6. Describe a dilation that uses a labeled point as its center and that would take F to H .

  7. Using B as the center of dilation, dilate H so that its image is itself. What scale factor did you use?

3.3 Getting Perspective

Follow the directions to perform the dilations in the applet.

  1. Dilate P  using C as the center and a scale factor of 4.
    1. Select the Dilate From Point tool. 
    2. Click on the object to dilate, and then click on the center of dilation.
    3. When the dialog box opens, enter the scale factor. Fractions can be written with plain text, ex. 1/2.
    4. Click
        
    5. Use the Ray tool and the Distance tool to verify.
  1. Dilate Q using C as the center and a scale factor of \frac12 .
open applet in presentation mode
  1. Draw a simple polygon. Choose a point not on the polygon to use as the center of dilation. Label it.
    1. Using your center point and a scale factor your teacher gives you, draw the image under the dilation of each vertex of the polygon, one at a time. Connect the dilated vertices to create the dilated polygon.
    2. Draw segments that connect each of the original vertices with its image. This will make your diagram look like a cool three-dimensional drawing of a box! If there's time, you can shade the sides of the box to make it look more realistic.
    3. Compare your drawing to other people’s drawings. What is the same and what is different? How do the choices you made affect the final drawing? Was your dilated polygon closer to your center point than to the original, or farther away? How is that determined?
open applet in presentation mode

Are you ready for more?

Here is line segment DE and its image D’E’  under a dilation.

Line segment ED is shown and a smaller line segment E'D' is shown next to it
  1. Use a ruler to find and draw the center of dilation. Label it F .
  2. What is the scale factor of the dilation?

Lesson 3 Summary

If A is the center of dilation, how can we find which point is the dilation of B with scale factor 2?

Points A, B, C, and D are shown on a ray.
Since the scale factor is larger than 1, the point must be farther away from A than B is, which makes C the point we are looking for. If we measure the distance between A and C , we would find that it is exactly twice the distance between A and B .

A dilation with scale factor less than 1 brings points closer. The point D is the dilation of B with center A and scale factor \frac{1}{3} .

Lesson 3 Practice Problems

  1. Segment AB measures 3 cm. Point O is the center of dilation. How long is the image of AB after a dilation with . . .

    1. Scale factor 5?
    2. Scale factor 3.7?
    3. Scale factor \frac 1 5 ?
    4. Scale factor s ?

  2. Here are points A and B . Plot the points for each dilation described.

    Points A and B are shown.
    1. C is the image of B using A as the center of dilation and a scale factor of 2.
    2. D is the image of A using B as the center of dilation and a scale factor of 2.
    3. E is the image of B using A as the center of dilation and a scale factor of \frac 1 2 .
    4. F is the image of A using B as the center of dilation and a scale factor of \frac 1 2 .
  3. Make a perspective drawing. Include in your work the center of dilation, the shape you dilate, and the scale factor you use.

  4. Triangle ABC is a scaled copy of triangle DEF . Side AB measures 12 cm and is the longest side of ABC . Side DE measures 8 cm and is the longest side of DEF .

    1. Triangle ABC is a scaled copy of triangle DEF with what scale factor?
    2. Triangle DEF is a scaled copy of triangle ABC with what scale factor?

  5. The diagram shows two intersecting lines.

    Find the missing angle measures.

    A circle is cut into 4 parts by intersecting lines. One angle created is 102 degrees. Angle B is the same as the found angle. Angles A and C are the same but undetermined.
    1. Show that the two triangles are congruent.
    2. Find the side lengths of DEF and the angle measures of ABC .
    Triangle ABC is shown on a graph with side lengths of 3, 5, and 3.2. Triangle DEF is shown with angles of 34.7, 108.4, and 36.9 degrees.