Sign In Sign Up Ask an AI advisor
fhbuvgy7836
fhbuvgy7836
15.12.2021 • 
Mathematics

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 and step 6 are incorrect. What is wrong with step 3 and what is wrong with step 6?

Tyler's Proof is below,

Step 1: Let
A
B
C
D
and
W
X
Y
Z
be rhombuses. By the definition of a rhombus
A
B

≅

B
C

≅

C
D

≅

D
A
, and
W
X

≅

X
Y

≅

Y
Z

≅

Z
W
.

Step 2: Dilate rhombus
A
B
C
D
by the scale factor given by
W
X
A
B
. Side
A
′
B
′
will be the same length as
W
X
because of how I chose the scale factor. Because all sides of a rhombus are congruent,
B
′
C
′
will be the same length as
W
X
and therefore the same length as
X
Y
,
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
X
Y
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
A
B
C
D
onto
W
X
Y
Z
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
W
X
coincide.

Step 6: If needed, reflect
A
"
′
B
"
′
C
"
′
D
"
′
over segment
W
X
so that
D
""
and
Z
are on the same side of
W
X
. Now segments
A
""
D
""
and
W
Z
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 and step 6 are incorrect. What is wrong with step 3 and what is wrong with step 6? Complete the explanations using the dropdown options.

Step 3 is incorrect because
the figures are actually congruent.
.

Step 6 is incorrect because
.

Solved
Show answers

Ask an AI advisor a question

I want advice