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Are these rectangles similar? Explain your reasoning.
Rectangles A B C D labeled Q and rectangle E F G H labeled R drawn on a square grid. Length of A B is 1, B C is 3, C D is 1, A D is 3, E F is 1, F G is 5, G H is 1, and E H is 5.
Tyler wrote a proof that all rectangles are similar. Make the image Tyler describes in each step in his proof. Which step makes a false assumption? Why is it false?
Step 1. Draw 2 rectangles. Label one and the other .
Step 2. Translate rectangle by the directed line segment from to . and now coincide. The points coincide because that’s how we defined our translation.
Step 3. Rotate rectangle , using as the center, so that is along ray .
Step 4. Dilate rectangle , using center and a scale factor of . Segments and now coincide. The segments coincide because was the center of the rotation, so and don’t move, and because and are on the same ray from , when we dilate by the right scale factor, it will stay on ray but be the same distance from as is, so and will coincide.
Step 5. Because all angles of a rectangle are right angles, segment now lies on ray . This is because the rays are on the same side of and make the same angle with it. (If and don’t coincide, reflect across so that the rays are on the same side of .)
Step 6. Dilate rectangle , using center and a scale factor of . Segments and now coincide by the same reasoning as in step 4.
Step 7. Due to the symmetry of a rectangle, if 2 rectangles coincide on 2 sides, they must coincide on all sides.
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. Consider any two circles, one centered at , with a radius of length , and the other centered at , with a radius of length . Translate the circle centered at along directed line segment .
Now dilate the image using center and a scale factor of . The circles should now coincide, proving that these circles are similar. Because any two circles could be described by the initial statement, this proves that any two circles are similar.
We can also show that all equilateral triangles are similar. Because we are talking about triangles, we can use the theorem that having all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion is enough to prove that the triangles are similar. All the pairs of corresponding angles are congruent because all the angles in any equilateral triangle measure . All the pairs of corresponding side lengths must be in the same proportion, because within each triangle, all the sides are congruent. Therefore, whatever scale factor works for one pair of sides will work for all 3 pairs of corresponding sides. If all pairs of corresponding sides are in the same proportion and all pairs of corresponding angles are congruent, then all equilateral triangles are similar.