# Are They All Similar?

ID: jokib-bofut
Illustrative Mathematics, CC BY 4.0
Subject: Geometry
Standards: HSG-SRT.A.2HSG-SRT.A.1HSG-SRT.C.6HSG-C.A.1

13 questions

# Are They All Similar?

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##### Stretched or Distorted? Rectangles

1) Are these rectangles similar?

True or false? Write below.

2) Explain how you know.

##### Faulty Logic

Tyler wrote a proof that all rectangles are similar:

1. Draw 2 rectangles. Label one ﻿$ABCD$﻿ and the other ﻿$PQRS$﻿.

2. Translate rectangle ﻿$ABCD$﻿ by the directed line segment from ﻿$A$﻿ to ﻿$P$﻿. ﻿$A'$﻿ and ﻿$P$﻿ now coincide. The points coincide because that’s how we defined our translation.

3. Rotate rectangle ﻿$A'B'C'D'$﻿ by angle ﻿$D'A'S$﻿. Segment ﻿$A''D''$﻿ now lies on ray ﻿$PS$﻿. The rays coincide because that’s how we defined our rotation.

4. Dilate rectangle ﻿$A''B''C''D''$﻿ using center ﻿$A''$﻿ and scale factor ﻿$\frac{PS}{AD}$﻿. Segments ﻿$A'''D'''$﻿ and ﻿$PS$﻿ now coincide. The segments coincide because ﻿$A''$﻿ was the center of the rotation, so ﻿$A''$﻿ and ﻿$P$﻿ don’t move, and since ﻿$D''$﻿ and ﻿$S$﻿ are on the same ray from ﻿$A''$﻿, when we dilate ﻿$D''$﻿ by the right scale factor, it will stay on ray ﻿$PS$﻿ but be the same distance from ﻿$A''$﻿ as ﻿$S$﻿ is, so ﻿$S$﻿ and ﻿$D'''$﻿ will coincide.

5. Because all angles of a rectangle are right angles, segment ﻿$A'''B'''$﻿ now lies on ray ﻿$PQ$﻿. This is because the rays are on the same side of ﻿$PS$﻿ and make the same angle with it. (If ﻿$A'''B'''$﻿ and ﻿$PQ$﻿ don’t coincide, reflect across ﻿$PS$﻿ so that the rays are on the same side of ﻿$PS$﻿.)

6. Dilate rectangle ﻿$A'''B'''C'''D'''$﻿ using center ﻿$A'''$﻿ and scale factor ﻿$\frac{PQ}{AB}$﻿. Segments ﻿$A'''B'''$﻿ and ﻿$PQ$﻿ now coincide by the same reasoning as in step 4.

7. Due to the symmetry of a rectangle, if 2 rectangles coincide on 2 sides, they must coincide on all sides.

3) Make the image Tyler describes in each step in his proof.

4) Identify the false assumption.

a)

1

b)

2

c)

3

d)

4

e)

5

f)

6

g)

7

5) Explain why the assumption is false.

##### Always? Prove it!

Read each statement and decide whether it is true or false.

6) All equilateral triangles are similar.

True or false? Write below.

7) If the statement is true, write a proof. If it is not, provide a counterexample.

8) All isosceles triangles are similar.

True or false? Write below.

9) If the statement is true, write a proof. If it is not, provide a counterexample.

10) All right triangles are similar.

True or false? Write below.

11) If the statement is true, write a proof. If it is not, provide a counterexample.

12) All circles are similar.

True or false? Write below.

13) If the statement is true, write a proof. If it is not, provide a counterexample.