Question

1. quadrilateral pqrs is dilated with the origin as the center of dilation to create quadrilateral pqrs. the coordinates of each vertex are integers.

which statement is true?

• each side length of quadrilateral pqrs is 1/2 the corresponding side length of quadrilateral pqrs.
• quadrilateral pqrs is congruent to quadrilateral pqrs.
• each angle measure of quadrilateral pqrs is 1/2 the corresponding angle measure of quadrilateral pqrs.
• quadrilateral pqrs is similar to quadrilateral pqrs.

Explanation:

Step1: Recall Dilation Properties

Dilation is a transformation that scales a figure. For a dilation with center at the origin, the image (P'Q'R'S') and pre - image (PQRS) have the following properties:

• Corresponding angles are congruent (angle measure remains the same).
• Corresponding sides are proportional (the ratio of side lengths of the image to the pre - image is equal to the scale factor).
• If the scale factor \(k\) is between 0 and 1, the image is a reduction of the pre - image; if \(k>1\), it is an enlargement. Also, dilated figures are similar (same shape, proportional sides, congruent angles).

Step2: Analyze Each Option

• Option 1: Since P'Q'R'S' is the image of PQRS under dilation (and from the graph, PQRS is larger than P'Q'R'S'), the side length of P'Q'R'S' is \(\frac{1}{2}\) of PQRS, not the other way around. So this option is wrong.
• Option 2: Congruent figures have the same size and shape. Since dilation changes the size (unless scale factor \(k = 1\)), P'Q'R'S' and PQRS are not congruent. This option is wrong.
• Option 3: Angles in similar figures (formed by dilation) are congruent, not scaled by a factor. So this option is wrong.
• Option 4: By the property of dilation, the image (P'Q'R'S') and the pre - image (PQRS) are similar. They have the same shape (corresponding angles congruent, corresponding sides proportional). This option is correct.

Answer:

Quadrilateral \(P'Q'R'S'\) is similar to quadrilateral \(PQRS\) (the last option among the given choices).