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:

To determine if rectangle \( EFGH \) is a dilation of rectangle \( ABCD \) with center at the origin, we analyze dilation properties. Dilation preserves parallelism and scales lengths by a constant ratio.

• For the first option: Check side lengths. Let's find coordinates. Assume \( ABCD \) has length (horizontal) from \( x=-3 \) to \( x = 3 \) (length \( 6 \))? Wait, looking at the grid, \( ABCD \): let's say \( A(-3,2) \), \( B(3,2) \), \( C(3,0) \), \( D(-3,0) \)? Wait, no, the inner rectangle \( ABCD \) and outer \( EFGH \). Wait, \( EFGH \): \( E(-4,4) \), \( F(4,4) \), \( G(4,0) \), \( H(-4,0) \)? Wait, no, the grid: \( E \) at \( x=-4, y=4 \), \( F \) at \( x=4, y=4 \), \( G \) at \( x=4, y=0 \), \( H \) at \( x=-4, y=0 \). \( ABCD \): \( A(-3,2) \), \( B(3,2) \), \( C(3,0) \), \( D(-3,0) \)? Wait, no, maybe \( ABCD \) has length \( 6 \) (from \( x=-3 \) to \( 3 \)) and \( EFGH \) has length \( 8 \) (from \( x=-4 \) to \( 4 \)). So ratio of lengths \( \frac{8}{6}=\frac{4}{3} \). Also, corresponding sides are parallel (horizontal and vertical sides, slopes 0 or undefined, same for both rectangles).

• Second option: Not all rectangles are similar (e.g., a 2x3 and 4x5 rectangle are not similar). So this is wrong.

• Third option: Center is origin? The lines from origin to corresponding vertices should be colinear. Let's check origin to \( A \) and \( E \). If \( A \) is (-3, 2) and \( E \) is (-4, 4), wait no, maybe my coordinate assumption is wrong. Wait, looking at the grid, \( D \) is at (-3, 0), \( H \) at (-4, 0); \( C \) at (3, 0), \( G \) at (4, 0); \( B \) at (3, 3), \( F \) at (4, 4); \( A \) at (-3, 3), \( E \) at (-4, 4). Wait, then vector from origin to \( A \) is (-3,3), to \( E \) is (-4,4). The ratio of components: \( \frac{-4}{-3}=\frac{4}{3} \), \( \frac{4}{3}=\frac{4}{3} \). So same scale factor. So center is origin. So third option is wrong.

• Fourth option: Corresponding sides are horizontal (slope 0) or vertical (undefined slope), same for both rectangles. So slopes are same. So fourth option wrong.

Thus, first option is correct: Yes, because corresponding sides are parallel and have lengths in the ratio \( \frac{4}{3} \).

Answer:

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