Unit 3, Lesson 8

Are They All Similar?

Geometry

Practice

Problem 1

This is an incorrect proof that all isosceles triangles are similar. Explain why the proof is incorrect.

Step 1. Draw 2 isosceles triangles \(ABC\) and \(DEF\) where \(AC=BC\) and \(DF=EF\).

Step 2. Dilate triangle \(ABC\) to a new triangle \(A’B’C\) using center \(C\) and a scale factor of \(\frac{DF}{AC}\), so that \(A’C=B’C=DF=EF\).

Step 3. Translate by directed line segment \(CF\) to take \(A’B’C\) to a new triangle \(A’’B’’F\). Because translation preserves distance, \(A’’F=A’C=DF\) and \(B’’F=B’C=EF\).

Step 4. Because \(A’’F=DF\), we can rotate using center \(F\) to take \(A’’\) to \(D\).

Step 5. Because \(B’’F=EF\), we can rotate using center \(F\) to take \(B’’\) to \(E\).

Step 6. We have now established a sequence of dilations, translations, and rotations that takes \(A\) to \(D\), \(B\) to \(E\), and \(C\) to \(F\), so the triangles are similar.