# Are They All Similar?

13 questions

# Are They All Similar?

##### Stretched or Distorted? Rectangles

1) Are these rectangles similar?

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.

1

2

3

4

5

6

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.

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

8) All isosceles triangles are similar.

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

10) All right triangles are similar.

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

12) All circles are similar.

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