Question

1. multiple choice:hans teacher asked him to draw a polygon similar to polygon a. here is his work. which explanation shows that polygons a and c are similar to each other? a. dilate polygon c with center at the upper - right corner and a scale factor of 3. since polygon c can be taken to polygon a with a dilation, the polygons are similar. b. since each side in polygon c is three times as long as the corresponding side in polygon a, the figures are similar. c. dilate polygon c with center at the upper - left vertex and a scale factor of \\( \frac{1}{3} \\). then, translate 7 units to the left. since polygon c can be taken to polygon a with a dilation followed by a translation, the figures are similar.

Explanation:

Step1: Analyze option a

Dilation with a scale factor of 3 from the upper - right corner. But when we check the side - length ratios, if we assume a side of polygon A has length \(x\), and a corresponding side of polygon C has length \(y\). If we consider the vertical side of polygon A (say length \(2\) units) and the vertical side of polygon C (say length \(6\) units), the scale factor is \(3\). However, dilation should be about a center. If we take the upper - right corner as the center, the relative positions of the non - adjacent vertices will not match the similarity transformation rules.

Step2: Analyze option b

If each side of polygon C is three times as long as the corresponding side of polygon A, by the definition of similar polygons (if the ratio of corresponding side lengths is constant and the corresponding angles are equal). For two polygons, if \(\frac{\text{side length of }C}{\text{side length of }A}=k\) (constant) and angles are equal (since dilation preserves angles), then they are similar.

Step3: Analyze option c

Dilation with a scale factor of \(\frac{1}{3}\) from the upper - left vertex of C to get A. But if we check the side - length ratios (e.g., vertical side of A is \(2\) and vertical side of C is \(6\)), the scale factor from C to A should be \(\frac{1}{3}\), but translation after dilation: If we first dilate C with scale factor \(\frac{1}{3}\) about the upper - left vertex, the position of the dilated figure (before translation) will not be in the position of A.

Answer:

B. Since each side in polygon C is three times as long as the corresponding side in polygon A, the figures are similar.