In previous lessons, students looked at the relationship between a figure and a scaled copy by finding the scale factor that relates the side lengths and by using tracing paper to compare the angles. This lesson takes both of these comparisons a step further.
- Students study corresponding distances between points that are not connected by segments, in both scaled and unscaled copies. They notice that when a figure is a scaled copy of another, corresponding distances that are not connected by a segment are also related by the same scale factor as corresponding sides.
- Students use protractors to test their observations about corresponding angles. They verify in several sets of examples that corresponding angles in a figure and its scaled copies are the same size.
Students use both insights—about angles and distances between points—to make a case for whether a figure is or is not a scaled copy of another (MP3). Practice with the use of protractors will help develop a sense for measurement accuracy, and how to draw conclusions from said measurements, when determining whether or not two angles are the same.
- Explain (orally and in writing) that corresponding angles in a figure and its scaled copies have the same measure.
- Identify (orally and in writing) corresponding distances or angles that can show that a figure is not a scaled copy of another.
- Recognize that corresponding distances in a figure and its scaled copy are related by the same scale factor as corresponding sides.
Let’s find relationships between scaled copies.
Make sure students have access to their geometry toolkits, especially rulers and protractors.
- I can use corresponding distances and corresponding angles to tell whether one figure is a scaled copy of another.
- When I see a figure and its scaled copy, I can explain what is true about corresponding angles.
- When I see a figure and its scaled copy, I can explain what is true about corresponding distances.
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).
A scaled copy is a copy of a figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
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