Lesson 2: Corresponding Parts and Scale Factors

Let’s describe features of scaled copies.

2.1: Number Talk: Multiplying by a Unit Fraction

Find each product mentally.

$\frac14 \boldcdot 32$

$(7.2) \boldcdot \frac19$

$\frac14 \boldcdot (5.6)$

2.2: Corresponding Parts

One road sign for railroad crossings is a circle with a large X in the middle and two R’s—with one on each side. Here is a picture with some points labeled and two copies of the picture. Drag and turn the moveable angle tool to compare the angles in the copies with the angles in the original.

GeoGebra Applet zHN6hAGH

  1. Complete this table to show corresponding parts in the three pictures.
    row 1 original Copy 1 Copy 2
    row 2 point $L$    
    row 3 segment $LM$    
    row 4   segment $ED$  
    row 5     point $X$
    row 6 angle $KLM$    
    row 7     angle $XYZ$
  2. Is either copy a scaled copy of the original road sign? Explain your reasoning.
  3. Use the moveable angle tool to compare angle $KLM$ with its corresponding angles in Copy 1 and Copy 2. What do you notice?
  4. Use the moveable angle tool to compare angle $NOP$ with its corresponding angles in Copy 1 and Copy 2. What do you notice?

2.3: Scaled Triangles

Here is Triangle O, followed by a number of other triangles.

Right triangle O, has sides 3, 4, 5. Right triangle A has sides 2, 3 halves, 5 halves. B has sides 6.08 and 6.32. C has sides 6, 7, 8. Right triangle D has sides 2, 5, and 5.39. Right triangle E has sides 2, 2, and 2.38. Right triangle F has sides 6, 8, and 10. Right triangle G has sides 3, 4, and 5. Right triangle H has sides 2, 8 thirds, and 10 thirds.

Your teacher will assign you two of the triangles to look at.

  1. For each of your assigned triangles, is it a scaled copy of Triangle O? Be prepared to explain your reasoning.
  2. As a group, identify all the scaled copies of Triangle O in the collection. Discuss your thinking. If you disagree, work to reach an agreement.
  3. List all the triangles that are scaled copies in the table. Record the side lengths that correspond to the side lengths of Triangle O listed in each column.
      Triangle O   3     4     5  
    row 1        
    row 2        
    row 3        
    row 4        
  4. Explain or show how each copy has been scaled from the original (Triangle O).

Summary

A figure and its scaled copy have corresponding parts, or parts that are in the same position in relation to the rest of each figure. These parts could be points, segments, or angles. For example, Polygon 2 is a scaled copy of Polygon 1.

Polygon 1 is A, F, E, D, C, B with side lengths, 2.8, 2, 1.3, 2, 3.2, and 3. Polygon 2 is G, L, K, J, I, H with side lengths 5.6, 4, 2.6, 4, 6.4, and 6.

  • Each point in Polygon 1 has a corresponding point in Polygon 2.
    For example, point $B$ corresponds to point $H$ and point $C$ corresponds to point $I$.

  • Each segment in Polygon 1 has a corresponding segment in Polygon 2.
    For example, segment $AF$ corresponds to segment $GL$.

  • Each angle in Polygon 1 also has a corresponding angle in Polygon 2.
    For example, angle $DEF$ corresponds to angle $JKL$.

The scale factor between Polygon 1 and Polygon 2 is 2, because all of the lengths in Polygon 2 are 2 times the corresponding lengths in Polygon 1. The angle measures in Polygon 2 are the same as the corresponding angle measures in Polygon 1: for example, the measure of angle $JKL$ is the same as the measure of angle $DEF$.

Practice Problems ▶

Glossary

scale factor

scale factor

The scale factor is the factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.

corresponding

corresponding

If a part of the original figure matches up with a part of the copy, we call them corresponding parts. The part could be an angle, point, or side, and you can have corresponding angles, corresponding points, or corresponding sides.

If you have a distance between two points in the original figure, then the distance between the corresponding points in the copy is called the corresponding distance.