After a dilation from the origin, ∆ABC maps onto ∆DEF. Which of the following statements is true?

Question

After a dilation from the origin, ∆ABC maps onto ∆DEF. Which of the following statements is true?

1.  m∠C = 4 · m∠F
2.  
3.  ∠E ≅ ∠B
4.  

Answer 1

After a dilation from the origin, ∆ABC maps onto ∆DEF. Which of the following statements is true?

Of the given alternatives, only the third, ∠E ≅ ∠B, is always true. Under dilation, corresponding angles (i.e., angles that occupy the same relative position on both the pre-image and the post-image) remain congruent. Here, ∠E and ∠B occupy the same relative position within their respective triangles, so it is true that they will be congruent under dilation.

We can rule out the other options as follows:

1. m∠C = 4 · m∠F is not true because ∠C and ∠F are corresponding angles, and corresponding angles remain congruent under dilation. But ∠F ≅ ∠C requires that m∠C = m∠F. So m∠C = 4 · m∠F can only be true if m∠C = m∠F = 0, in which case we don't actually have triangles.
2. is not true because the lengths of corresponding sides are proportional, not equal, under dilation. This statement is only true when the scaling factor equals 1.
3. is not true because BC and AC are sides of the same triangle, ∆ABC. The relationship of their lengths is unaffected under dilation. This statement is only true for a very specific value of the scaling factor, specifically .