Lesson 4: Scaled Relationships

Let’s find relationships between scaled copies.

4.1: Three Quadrilaterals (Part 1)

Each of these polygons is a scaled copy of the others.

Polygon A, B, C, D. Polygon E, F, G, H. Polygon I, J, K, L.
  1. Name two pairs of corresponding angles. What can you say about the sizes of these angles?
  2. Check your prediction by measuring at least one pair of corresponding angles using a protractor. Record your measurements to the nearest $5^\circ$.

4.2: Three Quadrilaterals (Part 2)

Each of these polygons is a scaled copy of the others. You already checked their corresponding angles.

  1. The side lengths of the polygons are hard to tell from the grid, but there are other corresponding distances that are easier to compare. Identify the distances in the other two polygons that correspond to $DB$ and $AC$, and record them in the table.
    quadrilateral distance that
    corresponds to $DB$    		 distance that
    corresponds to $AC$
    $ABCD$ $DB = 4$ $AC = 6$
    $EFGH$    
    $IJKL$    

  2. Look at the values in the table. What do you notice?

    Pause here so your teacher can review your work.

  3. The larger figure is a scaled copy of the smaller figure.

    Both figures resemble the letter W. Tracing each, the smaller figure's vertices in order are A, B, C, D, E. The length of segment A, B is 6. The larger W is H, I, J, K, L. The length of segment H, I is 15.

    1. If $AE = 4$, how long is the corresponding distance in the second figure? Explain or show your reasoning.
    2. If $IK = 5$, how long is the corresponding distance in the first figure? Explain or show your reasoning.

4.3: Scaled or Not Scaled?

Here are two quadrilaterals.

Two quadrilaterals on a coordinate plane. The first figure is labeled JXNY. Point X is 2 units to the left and 8 units up from point J. Point N is 2 units to the right and 1 unit up from point X. Point Y is 4 units to the right and 1 unit down from point N. Point J is 4 units to the left and 8 units down from point Y. Point N is directly above point J. The second figure is labeled ZHCS. Point Z is 1 unit to the left and 5 units up from point S. Point H is 1 unit to the right and 1 unit up from point Z. Point C is 3 units to the right and 1 unit down from point H. Point S is 3 units to the left and 5 units down from point C. Point H is directly above point S.
  1. Mai says that Polygon $ZSCH$ is a scaled copy of Polygon $XJYN$, but Noah disagrees. Do you agree with either of them? Explain or show your reasoning.
  2. Record the corresponding distances in the table. What do you notice?
    quadrilateral horizontal distance vertical distance
    $XJYN$ $XY = \phantom{33}$ $JN = \phantom{33}$
    $ZSCH$ $ZC = \phantom{33}$ $SH = \phantom{33}$
  3. Measure at least three pairs of corresponding angles in $XJYN$ and $ZSCH$ using a protractor. Record your measurements to the
    nearest $5^\circ$. What do you notice?
  4. Do these results change your answer to the first question? Explain.

Here are two more quadrilaterals.

The angle measures, in degrees, for both trapezoids are: 60, 60, 120, 120. In A, B, C, D, the top length is 2, bottom length is 6, both sides lengths are 4. In E, F, G, H, the top length is 1, bottom length is 4 and both side lengths are 3.
  1. Kiran says that Polygon $EFGH$ is a scaled copy of $ABCD$, but Lin disagrees. Do you agree with either of them? Explain or show your reasoning.

4.4: Comparing Pictures of Birds

Here are two pictures of a bird. Find evidence that one picture is not a scaled copy of the other. Be prepared to explain your reasoning.

GeoGebra Applet TAPZhcqg

Summary

When a figure is a scaled copy of another figure, we know that:

  1. All distances in the copy can be found by multiplying the corresponding distances in the original figure by the same scale factor, whether or not the endpoints are connected by a segment.

    For example, Polygon $STUVWX$ is a scaled copy of Polygon $ABCDEF$. The scale factor is 3. The distance from $T$ to $X$ is 6, which is three times the distance from $B$ to $F$.

  1. All angles in the copy have the same measure as the corresponding angles in the original figure, as in these triangles.

Original triangle has angle measures 42, 60, and 78 degrees. The larger, scaled version of the triangle has angle measures 42, 60, and 78 degrees.

These observations can help explain why one figure is not a scaled copy of another.

For example, even though their corresponding angles have the same measure, the second rectangle is not a scaled copy of the first rectangle, because different pairs of corresponding lengths have different scale factors, $2 \boldcdot \frac12 = 1$ but $3 \boldcdot \frac23 = 2$.

The first rectangle has height 2 and length 3. The second rectangle has height 1 and length 2.

Practice Problems ▶