Question

Which transformations could be performed to show that △ABC is similar to △A ′ B′C' ?

a. A reflection over the x-axis, then a dilation by a scale factor of 3
b. A reflection over the x-axis, then a dilation by a scale factor of 6
c. A 180° rotation about the origin, then a dilation by a scale factor of 3
d. A 180° rotation about the origin, then a dilation by a scale factor of 6

Answer 1

Transformations like reflections, rotations, and dilations can be utilized to demonstrate the similarity between triangles △ABC and △A′B′C′. The correct combination of a reflection or rotation followed by a dilation with the right scale factor (either 3 or 6) will depend on the actual sizes of the triangles.

Step-by-step explanation:

A reflection over the x-axis followed by a dilation would preserve the angle measures and create proportional side lengths if the correct scale factor is applied. A reflection does not change the size of the triangle, just its orientation, so the next step should ensure that the triangles are properly scaled.

Whether we choose options (a) or (b) depends on the relationship between △ABC and △A′B′C′. If △A′B′C′ is three times larger than △ABC, then a reflection over the x-axis followed by a dilation by a scale factor of 3 will show that these triangles are similar. If △A′B′C′ is six times larger, the scale factor should be 6 instead. Options (c) and (d) also involve a rotation which is an isometry and preserves similarity, followed by a dilation.