Question

is rectangle efgh the result of a dilation of rectangle abcd with a center of dilation at the origin? why or why not? yes, because corresponding sides are parallel and have lengths in the ratio \\(\frac{4}{3}\\). yes, because both figures are rectangles and all rectangles are similar. no, because the center of dilation is not at (0, 0). no, because corresponding sides have different slopes.

Explanation:

Step1: Recall Dilation Properties

Dilation requires corresponding sides to be parallel (since it's a similarity transformation) and the ratio of corresponding side lengths to be constant (scale factor). Also, the center of dilation is the origin here, so we check side lengths and parallelism.

Step2: Analyze Each Option

• Option 1: Check side lengths. Let's find coordinates. For ABCD: Let's say A(-3,2), B(3,2), C(3,0), D(-3,0)? Wait, no, looking at grid: ABCD: D(-3,0), C(3,0)? Wait, EFGH: H(-4,0), G(4,0), F(4,2), E(-4,2)? Wait, ABCD: D(-3,0), C(3,0), B(3,2), A(-3,2)? So length of ABCD's base: from -3 to 3, so 6 units? Wait, no, x-coordinates: D is at -3? Wait, grid lines: D is at x=-3? Wait, EFGH: H is at x=-4, G at x=4. So length of EFGH's base: 8 units. ABCD's base: from -3 to 3, so 6 units? Wait, no, maybe ABCD is from -2 to 2? Wait, maybe I misread. Wait, the inner rectangle ABCD: D is at x=-3? No, the grid has x-axis with -4, -2, 0, 2, 4. So EFGH: H(-4,0), G(4,0), F(4,2), E(-4,2). So length EH to FG: 8 units (from -4 to 4). ABCD: D(-3,0), C(3,0), B(3,2), A(-3,2). So length DC to AB: 6 units (from -3 to 3). Wait, 8/6 = 4/3. And corresponding sides are horizontal (slope 0) and vertical (undefined slope), so parallel. So option 1 says corresponding sides are parallel (they are, both rectangles, so sides are horizontal/vertical, parallel) and ratio 4/3 (8/6=4/3).

• Option 2: All rectangles are similar? No, rectangles have different aspect ratios (length/width). For example, a 2x3 rectangle and 4x5 are not similar. So this is wrong.

• Option 3: Center of dilation is origin? The problem says center at origin, so this is wrong.

• Option 4: Corresponding sides have different slopes? Both rectangles have horizontal (slope 0) and vertical (undefined) sides, so slopes are same. So wrong.

So Option 1 is correct.

Answer:

A. Yes, because corresponding sides are parallel and have lengths in the ratio $\frac{4}{3}$.