Question

1. becca performed two transformations on triangle abc to create triangle a\b\c\, but she cant remember the order she performed them. she either rotated the triangle 90° clockwise about the origin and then dilated it by a scale factor of 1/2 with a center of dilation at the origin, or she performed the transformations in the opposite order.

which order did becca perform the transformations? choose one option from each drop - down menu to answer the question.
becca choose...
if she must have rotated the figure first
must have dilated the figure first
could have performed the transformations in either order
rotated triangle abc and then dilated it, the resulting triangle would be choose...
on the coordinate grid. if she dilated a different size than triangle a\b\c\
in a different location than triangle a\b\c\
choose...
on the coordinate grid.
choose...
triangle a\b\c\ as it appears
a different size than triangle a\b\c\
in a different location than triangle a\b\c\

Explanation:

Step1: Recall transformation rules

The rules for a 90 - degree clock - wise rotation about the origin are $(x,y)\to(y, - x)$ and for dilation by a scale factor $k=\frac{1}{2}$ about the origin are $(x,y)\to(\frac{1}{2}x,\frac{1}{2}y)$.

Step2: Analyze order of transformations

If we rotate first: Let a point $(x,y)$ on $\triangle ABC$ be rotated 90 - degree clock - wise to $(y, - x)$ and then dilated by $\frac{1}{2}$ to $(\frac{1}{2}y,-\frac{1}{2}x)$. If we dilate first: A point $(x,y)$ is dilated to $(\frac{1}{2}x,\frac{1}{2}y)$ and then rotated 90 - degree clock - wise to $(\frac{1}{2}y,-\frac{1}{2}x)$. The two - step transformations of rotation and dilation about the origin are commutative when the center of dilation is the origin.

Answer:

Becca could have performed the transformations in either order. If she rotated triangle $ABC$ and then dilated it, the resulting triangle would be in the same location as if she dilated first and then rotated.