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Step 3: If all the sides in 2 figures are proportional, then those 2 figures are similar, so there must be a sequence of transformations that talke ABCD onto WXYZ using dilations and rigid motions.

Step 3: If all the sides in 2 figures are proportional, then those 2 figures are similar, so there must be a sequence of transformations that talke ABCD onto WXYZ using dilations and rigid motions.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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What is wrong in step 3 

Tyler was trying to prove that all rhombuses are similar. He thought, "If I draw 2 rhombuses, I can find dilations that will take one rhombus exactly onto the other." Then he wrote a proof. All rhombuses are NOT similar. Tyler's proof has flaws. Step 3 is his first incorrect step. Explain what is wrong with step 3. Tyler's Proof is below, Step 1: Let ABCD and WXYZ be rhombuses. By the definition of a rhombus ABE BC CD DA, andWX = XY E YZ 2 ZW. Step 2: Dilate rhombus ABCD by the scale factor given by Side A'B will be the same length as WX because of how I chose the scale factor. Because all sides of a rhombus are congruent, B'C' will AB be the same length as WX and therefore the same length as XY, C'D will be the same length as YZ, and A'D will be the same length as WZ. That means all the corresponding sides of A'B'C'D' and WXYZ will be the same length. Step 3: If all the sides in 2 figures are proportional, then those 2 figures are similar, so there must be a sequence of transformations that take ABCD onto WXY Z using dilations and rigid motions. Step 4: Translate A'B'C'D' by the directed line segment A'W so that A and WV coincide. Step 5: Rotate A" B" C" D" by angle B" WX so that B and X coincide. Now segments A " B " and WX coincide. Step 6: If needed, reflect A B C D" over segment WX so that D" and Z are on the same side of WX. Now segments A"" D "" and WZ coincide. Step 7: Once 3 vertices of the rhombus are lined up, the other vertex has to line up as well or else the shapes wouldn't be rhombuses. So, we have shown we can use rigid motions and dilations to line up any 2 rhombuses, and therefore, all rhombuses are similar. Step 3 is his first incorrect step in Tyler's proof. Explain what is wrong with step 3. B IU E E !! INTL 1 6:21
Transcribed Image Text:Tyler was trying to prove that all rhombuses are similar. He thought, "If I draw 2 rhombuses, I can find dilations that will take one rhombus exactly onto the other." Then he wrote a proof. All rhombuses are NOT similar. Tyler's proof has flaws. Step 3 is his first incorrect step. Explain what is wrong with step 3. Tyler's Proof is below, Step 1: Let ABCD and WXYZ be rhombuses. By the definition of a rhombus ABE BC CD DA, andWX = XY E YZ 2 ZW. Step 2: Dilate rhombus ABCD by the scale factor given by Side A'B will be the same length as WX because of how I chose the scale factor. Because all sides of a rhombus are congruent, B'C' will AB be the same length as WX and therefore the same length as XY, C'D will be the same length as YZ, and A'D will be the same length as WZ. That means all the corresponding sides of A'B'C'D' and WXYZ will be the same length. Step 3: If all the sides in 2 figures are proportional, then those 2 figures are similar, so there must be a sequence of transformations that take ABCD onto WXY Z using dilations and rigid motions. Step 4: Translate A'B'C'D' by the directed line segment A'W so that A and WV coincide. Step 5: Rotate A" B" C" D" by angle B" WX so that B and X coincide. Now segments A " B " and WX coincide. Step 6: If needed, reflect A B C D" over segment WX so that D" and Z are on the same side of WX. Now segments A"" D "" and WZ coincide. Step 7: Once 3 vertices of the rhombus are lined up, the other vertex has to line up as well or else the shapes wouldn't be rhombuses. So, we have shown we can use rigid motions and dilations to line up any 2 rhombuses, and therefore, all rhombuses are similar. Step 3 is his first incorrect step in Tyler's proof. Explain what is wrong with step 3. B IU E E !! INTL 1 6:21
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