In Grade 8, students informally showed that a dilation maps a segment to a segment on the coordinate plane. The lesson includes an opening discussion that reminds students of this fact. Next, students must consider how to prove that dilations map segments to segments when the segment is not tied to the coordinate plane. We again call upon our knowledge of the triangle side splitter theorem to show that a dilation maps a segment to a segment. The goal of the lesson is for students to understand the effect that dilation has on segments, specifically that a dilation will map a segment to a segment so that its length is times the original.
How Do Dilations Map Segments?
a. Is a dilated segment still a segment? If the segment is transformed under a dilation, explain how.
b. Dilate the segment 𝑃𝑄 by a scale factor of 2 from center 𝑂.
i. Is the dilated segment 𝑃′𝑄′ a segment?
ii. How, if at all, has the segment 𝑃𝑄 been transformed?
Case 1. Consider the case where the scale factor of dilation is 𝑟 = 1. Does a dilation from center 𝑂 map segment 𝑃𝑄 to a segment 𝑃′𝑄′? Explain.
Case 2. Consider the case where a line 𝑃𝑄 contains the center of the dilation. Does a dilation from the center with scale factor 𝑟 ≠ 1 map the segment 𝑃𝑄 to a segment 𝑃′𝑄′? Explain.
Case 3. Consider the case where 𝑃𝑄 does not contain the center 𝑂 of the dilation, and the scale factor 𝑟 of the dilation is not equal to 1; then, we have the situation where the key points 𝑂, 𝑃, and 𝑄 form △ 𝑂𝑃𝑄. The scale factor not being equal to 1 means that we must consider scale factors such that 0 < 𝑟 < 1 and 𝑟 > 1. However, the proofs for each are similar, so we focus on the case when 0 < 𝑟 < 1. When we dilate points 𝑃 and 𝑄 from center 𝑂 by scale factor 0 < 𝑟 < 1, as shown, what do we know about points 𝑃′ and 𝑄′?
We use the fact that the line segment 𝑃′𝑄′ splits the sides of △ 𝑂𝑃𝑄 proportionally and that the lines containing 𝑃𝑄 and 𝑃′𝑄′ are parallel to prove that a dilation maps segments to segments. Because we already know what happens when points 𝑃 and 𝑄 are dilated, consider another point 𝑅 that is on the segment 𝑃𝑄. After dilating 𝑅 from center 𝑂 by scale factor 𝑟 to get the point 𝑅′, does 𝑅′ lie on the segment 𝑃′𝑄′?
Putting together the preliminary dilation theorem for segments with the dilation theorem, we get
DILATION THEOREM FOR SEGMENTS: A dilation 𝐷𝑂,𝑟 maps a line segment 𝑃𝑄 to a line segment 𝑃′𝑄′ sending the endpoints to the endpoints so that 𝑃′𝑄′ = 𝑟𝑃𝑄. Whenever the center 𝑂 does not lie in line 𝑃𝑄 and 𝑟 ≠ 1, we conclude 𝑃𝑄 || 𝑃′𝑄′. Whenever the center 𝑂 lies in 𝑃𝑄 or if 𝑟 = 1, we conclude 𝑃𝑄 = 𝑃′𝑄′. As an aside, observe that a dilation maps parallel line segments to parallel line segments. Further, a dilation maps a directed line segment to a directed line segment that points in the same direction.
Now look at the converse of the dilation theorem for segments: If 𝑃𝑄 and 𝑅𝑆 are line segments of different lengths in the plane, then there is a dilation that maps one to the other if and only if 𝑃𝑄 = 𝑅𝑆 or 𝑃𝑄 || 𝑅𝑆 . Based on Examples 2 and 3, we already know that a dilation maps a segment 𝑃𝑄 to another line segment, say 𝑅𝑆̅̅̅̅, so that 𝑃𝑄 = 𝑅𝑆 (Example 2) or 𝑃𝑄 || 𝑅𝑆 (Example 3). If 𝑃𝑄 || 𝑅𝑆 , then, because 𝑃𝑄 and 𝑅𝑆 are different lengths in the plane, they are bases of a trapezoid, as shown.
Since 𝑃𝑄 and 𝑅𝑆 are segments of different lengths, then the non-base sides of the trapezoid are not parallel, and the lines containing them meet at a point 𝑂 as shown.
Recall that we want to show that a dilation maps 𝑃𝑄 to 𝑅𝑆. Explain how to show it. The case when the segments 𝑃𝑄 and 𝑅𝑆 are such that 𝑃𝑄 = 𝑅𝑆 is left as an exercise.
In the following exercises, you will consider the case where the segment and its dilated image belong to the same line, that is, when 𝑃𝑄 and 𝑅𝑆 are such that 𝑃𝑄 = 𝑅𝑆 .
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