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22-23 Geometry Unit 3 EUA
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. Which steps in Tyler's proof have flaws? Explain what is wrong with each step that is wrong.
Step 1: Let A BCD and W XYZ be rhombuses. By the definition of a rhombus, AB = BOECDE DA and WXEXY AYZEZW.
Step 2: Dilate rhombus ABC D by the scale factor given by
WX
4B Side A'B' will be the same length as W X because of how I chose the scale factor. Because all sides ofa
thombus are congruent, B'C' will be the same length as W X and therefore the same length as XY, C'D' will be the same length as Y Z, and A'D' will be the same length as W Z. That means all the corresponding sides of A' B'C'D and W XY Z 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 ABC D onto W XYZ using dilations and rigid motions.
Step 4: Translate A' B'C'D' by the directed line segment A'W so that A" and W coincide.
Step 5: Rotate A" B"C" D" by angle B"W X so that B/' and X coincide. Now segments A' B'' and WX coincide.
Step 6: If needed, reflect A" B''O' D'' over segment W X so that D'' and Z are on the same side of WIX. Now segments A"' D' and WZ coincide.

Answer 1

Step-by-step explanation: Step 1: Tyler correctly states that rhombuses have all sides congruent.

Step 2: Tyler correctly states that dilating a rhombus by a scale factor will result in sides of the same length, but he does not provide a justification for why he chose the scale factor of WX/AB.

Step 3: Tyler correctly states that if all sides in two figures are proportional, then the figures are similar, but he does not provide a justification for why this is true.

Step 4: Tyler correctly states that he will translate A'B'C'D' by the directed line segment A'W, but he does not provide a justification for why this will result in A'' and W coinciding.

Step 5: Tyler correctly states that he will rotate A"B"C"D" by the angle B"WX, but he does not provide a justification for why this will result in B'' and X coinciding.

Step 6: Tyler correctly states that he will reflect A"B''C''D'' over segment WX if needed, but he does not provide a justification for why this will result in D'' and Z coinciding.