Question

figure mnop is plotted on a coordinate plane. it is rotated 180 degrees clockwise about the origin and then dilated by a scale factor of 0.5 to get image mnop.
which statement must be true of the two figures?
a the figures are congruent because of a reduction in the original figure.
b the figures are similar because of an enlargement in the original figure.
c the figures are congruent because of enlargement in the original figure.
d the figures are similar because of a reduction in the original figure.

Explanation:

Step1: Recall the properties of congruent and similar figures

Congruent figures have the same shape and size. Similar figures have the same shape but not necessarily the same size. A rotation is a rigid transformation (preserves shape and size). A dilation with a scale factor \(k = 0.5\) changes the size of the figure.

Step2: Analyze each option

• Option A: A reduction (dilation with \(0
• Option B: An enlargement (dilation with \(k>1\)) changes the size. Here, the scale factor is \(0.5\) (a reduction, not an enlargement). So, option B is incorrect.
• Option C: Congruent figures must have the same size. Since there is a dilation (scale - factor \(0.5\)), the size is changed. So, option C is incorrect.
• Option D: Similar figures have the same shape. A rotation (rigid transformation) preserves shape and a dilation (even with \(k = 0.5\)) preserves shape. So, the two figures are similar because of a reduction (dilation with \(k=0.5\)) in the original figure.

Answer:

D. The figures are similar because of a reduction in the original figure.