Question

Shape I is similar to Shape II. The sequence applied to Shape I that proves Shape I is similar to Shape II is a translation right and up, and then a dilation by a scale factor of ____. 1: 0.5 2: 1 3: 1.5 4: 2

Answer 1

The sequence that proves Shape I is similar to Shape II includes a translation and a dilation with a scale factor of 2, as indicated by the proportion 1:2=4:x, which solves to x=8. The correct scale factor for converting a scale dimension of 4 to an actual dimension of 8 is 1:2, meaning the object has been scaled up by a factor of 2.

Step-by-step explanation:

When considering the similarity of two shapes, it is important to apply a sequence of transformations to determine if one shape could be resized, repositioned, or rotated to become congruent to the other. In this case, Shape I undergoes a translation, followed by a dilation. To prove that Shape I is similar to Shape II, we must identify the correct scale factor that has been used.

Using the proportion given, 1:2=4:x, we can solve for the unknown x, which represents the actual dimension. By setting up a cross-multiplication, we get 1*x = 2*4, leading us to x = 8. Therefore, the scale factor to go from a scale dimension of 4 to an actual dimension of 8 is 1:2, which means that Shape II is twice the size of Shape I. It is clear from this logic that the dilation involved a scaling up by factor of 2.

In the example of Block B's height being twice that of Block A, this further demonstrates how changes in dimensions can be proportional based on scale factor. This is a foundational concept when determining actual dimensions using scale models.

The scale ratio, such as 2:1, indicates how to calculate the scale factor. In the case where 3 inches equates to 12 feet, the ratio would be 3 inches to 144 inches (since there are 12 inches in a foot), making the scale factor 1:48 or 1/48 after converting both measurements to the same units.

Answer 2

The scale factor for the dilation applied to Shape I to make it similar to Shape II is 1.5.

Explanation:

The given sequence of translation right and up indicates a movement in the positive x-direction and positive y-direction, aligning Shape I with Shape II. Following the translation, a dilation is performed, and the scale factor is determined by the change in size from Shape I to Shape II.

In this case, Shape I is enlarged by a factor of 1.5 to match the corresponding dimensions of Shape II. This dilation ensures that the proportional relationships between the corresponding sides of the two shapes are maintained.

A scale factor of 1 would mean no change in size, while a factor less than 1 would result in a reduction in size. Conversely, a scale factor greater than 1 indicates an enlargement. Therefore, a scale factor of 1.5 is the correct value to achieve similarity between Shape I and Shape II after the specified translation and dilation.