Question

7 multiple answer 4 points a. dilate by a scale factor of 2 with a center of dilation at the origin. reflect △abc across the x - axis. dilate by a scale factor of 1/2 with a center of dilation at the origin. reflect △abc across the y - axis. 8 multiple answer 4 points b. dilate by a factor of 2/3 with a center of dilation at the origin. rotate △abc about the origin 90 degrees counterclockwise. dilate by a factor of 1.5 with a center of dilation at the origin. rotate △abc about the origin 90 degrees clockwise.

Explanation:

Step1: Recall transformation rules

Dilation changes size, reflection flips over an axis, rotation turns around a point.

Step2: Analyze each option

We don't have the original and target triangles clearly shown, but we can still discuss each transformation in general terms for understanding.

• Dilation by a scale - factor of 2 with center at origin doubles the size of the figure.
• Reflecting $\triangle ABC$ across the $x$-axis changes the sign of the $y$-coordinates of its vertices ($(x,y)\to(x, - y)$).
• Dilation by a scale - factor of $\frac{1}{2}$ with center at origin halves the size of the figure.
• Reflecting $\triangle ABC$ across the $y$-axis changes the sign of the $x$-coordinates of its vertices ($(x,y)\to(-x,y)$).
• Dilation by a factor of $\frac{2}{3}$ with center at origin makes the figure smaller.
• Rotating $\triangle ABC$ 90 degrees counter - clockwise about the origin changes the coordinates as $(x,y)\to(-y,x)$.
• Dilation by a factor of 1.5 with center at origin makes the figure larger.
• Rotating $\triangle ABC$ 90 degrees clockwise about the origin changes the coordinates as $(x,y)\to(y, - x)$.

Since no specific requirements about the transformation from an original to a target triangle are given, we assume we are just identifying types of transformations.

Answer:

All of the options (a. Dilate by a scale factor of 2 with a center of dilation at the origin; Reflect $\triangle ABC$ across the $x$-axis; Dilate by a scale factor of $\frac{1}{2}$ with a center of dilation at the origin; Reflect $\triangle ABC$ across the $y$-axis; b. Dilate by a factor of $\frac{2}{3}$ with a center of dilation at the origin; Rotate $\triangle ABC$ about the origin 90 degrees counterclockwise; Dilate by a factor of 1.5 with a center of dilation at the origin; Rotate $\triangle ABC$ about the origin 90 degrees clockwise) are valid geometric transformations.