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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.
b
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'(0,0) B(2,4) & ⟶ & B'(4,-2) C(6,0) & ⟶ & C'(0,-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 congruent. Let's focus on the two transformations in which the image is congruent to the original triangle.
| Letter | Image | Transformation |
|---|---|---|
| b | Congruent | A 90^(∘) clockwise rotation about the origin |
| c | Congruent | A reflection across the x-axis |
Next, notice that only point A is unchanged after the transformation. In a reflection across the x-axis, all points on the x-axis are unchanged. Point C lies on the x-axis. However, point C moved, so our transformation cannot be a reflection. A 90^(∘) clockwise rotation about the origin is out best candidate.
| Letter | Image | Transformation |
|---|---|---|
| b | Congruent | A 90^(∘) clockwise rotation about the origin |
Let's check that this answer is correct. We can apply this transformation to the original triangle and to see if the results match.
Our transformation is correct!
