• Students verify the converse of the Fundamental Theorem of Similarity experimentally.
• Students apply the Fundamental Theorem of Similarity to find the location of dilated points on the plane.
Converse of the Fundamental Theorem of Similarity:
If lines PQ and P'Q' are parallel, and |P'Q'| = r|PQ| then from a center O, P'= dilation(P), Q' = dilation(Q), and |OP'| = r|OP| and |OQ'| = r|OQ|.
To find the coordinates of a dilated point, we must use what we know about FTS, dilation, and scale factor.
Discuss the Converse of the Fundamental Theorem of Similarity (FTS)
1. In the diagram below, points P and Q have been dilated from center O by scale factor r. PQ || P'Q', |PQ| = 5 cm, and |P'Q'| = 10 cm.
a. Determine the scale factor r.
b. Locate the center of dilation. Measure the segments to verify that |OP'| = r|OP| and |OQ'| = r|OQ|.
Show your work below.
• Now that we know FTS and the converse of FTS in terms of dilations, we will practice using them to find the coordinates of points and dilated points on a plane. We will begin simply.
• In the diagram we have center O and ray OA. We want to find the coordinates of point A'. We are given that the scale factor of dilation is r = 2.
• To find A' we could use a ruler or compass to measure |OA|, but now that we know about FTS, we can do this another way. First, we should look for parallel lines that will help us locate point A'. How can we use the coordinate plane to ensure parallel lines?
• Let’s use the x-axis as one of our rays. (Show picture below). Where should we place a point, B, on the ray along the x-axis?
• (Show picture below.) This is beginning to look like the activity we did in Lesson 4. We know that that scale factor r = 2. Where should we put point B'?
• (Show picture below.) Now that we know the location of B', using FTS, what do we expect to be true about the lines containing segments AB and A'B'?
• (Show picture below.) Then what is the location of point A'?
• (Show picture below.) Could point A' be located anywhere else? Specifically, could A' have a different y- coordinate? Why or why not?
• Could point A' be located at another location that has 10 as its x-coordinate? For example, could A' be at (10, 5)? Why or why not?
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