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: Check properties of dilation

Dilation is a transformation that enlarges or reduces a figure by a scale factor with respect to a center. For dilation about the origin, corresponding sides of the original and dilated figure are parallel and the ratio of their lengths is the scale factor.

Step2: Analyze each option

• Option 1:
• If it were a dilation, for example, if we assume side lengths. Let's assume some side - length calculations (by counting grid units). Suppose length of a side in \(ABCD\) and \(EFGH\). If we assume the ratio of lengths is \(\frac{4}{3}\) and sides are parallel (since rectangles have sides parallel to the axes), this option seems correct.
• Option 2:
• Not all rectangles are similar. Similarity requires that the ratio of corresponding side lengths is the same. For example, a rectangle with sides \(2\times3\) and a rectangle with sides \(2\times4\) are not similar. So, the statement “all rectangles are similar” is false.
• Option 3:
• The problem states “center of dilation at the origin”. So, this option is incorrect as it contradicts the problem's condition about the center of dilation.
• Option 4:
• Since \(ABCD\) and \(EFGH\) are rectangles (sides parallel to the \(x\) and \(y\) - axes), corresponding sides have the same slope (either \(0\) for horizontal sides or undefined for vertical sides in the case of non - algebraic slope, or we can use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For horizontal sides \(y_2=y_1\), so \(m = 0\); for vertical sides \(x_2=x_1\), and the slope is undefined in the algebraic sense, but they are parallel). So, this option is incorrect.

Answer:

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