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Draw the original triangle and the image triangle on the same coordinate plane. Check whether the triangles are congruent or only similar.
d
We are given the vertices of a triangle and told how it is transformed. Let's see the vertices of this triangle before and after the transformation. ccc Before & & After A(0,0) & ⟶ & A'(2, -6) B(2,4) & ⟶ & B'(6, 2) C(6,0) & ⟶ & C'(14, -6) We want to know which of the following four transformations produced the image triangle.
| Letter | Image | Transformation |
|---|---|---|
| a | Similar | A dilation with a scale factor of 12 |
| b | Congruent | A 90^(∘) clockwise rotation about the origin |
| c | Congruent | A reflection across the x-axis |
| d | Similar | A translation of (x+1, y-3) followed by a dilation with a scale of 2 |
The two triangles seem to be similar but not congruent. Let's focus on the two transformations in which the image is similar to the original triangle.
| Letter | Image | Transformation |
|---|---|---|
| a | Similar | A dilation with a scale factor of 12 |
| d | Similar | A translation of (x+1, y-3) followed by a dilation with a scale of 2 |
Next, notice that all of the vertices changed their positions. The position of a center of dilation does not change. When we dilate the given triangle, point A is the center of dilation because it is the origin and multiplication by 0 is always 0. Transformation a does not move point A so it cannot be the answer. Let's check the last candidate we have.
| Letter | Image | Transformation |
|---|---|---|
| d | Similar | A translation of (x+1, y-3) followed by a dilation with a scale of 2 |
Let's check if this answer is correct. We can apply this transformation to the original triangle to see if the results match.
Our transformation is correct!
