Are They All Similar?

ID: jokib-bofut
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
Standards: HSG-SRT.A.2HSG-SRT.A.1HSG-SRT.C.6HSG-C.A.1
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13 questions

Are They All Similar?

Classroom:
Due:
Student Name:
Date Submitted:
Stretched or Distorted? Rectangles


A template for answering this question. Ask your instructor for an alternative.

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 ABCDABCD and the other PQRSPQRS.

2. Translate rectangle ABCDABCD by the directed line segment from AA to PP. AA' and PP now coincide. The points coincide because that’s how we defined our translation.

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

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

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

6. Dilate rectangle ABCDA'''B'''C'''D''' using center AA''' and scale factor PQAB\frac{PQ}{AB}. Segments ABA'''B''' and PQPQ 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.