Question

lesson 3: measuring dilations
cool down: describing stretching
triangle ( p ^ { prime } q ^ { prime } r ^ { prime } ) is a dilation of triangle ( p q r ) using center ( c ) and scale factor 1.5.

1. is triangle ( p ^ { prime } q ^ { prime } r ^ { prime } ) larger or smaller than triangle ( p q r ) ? explain how you know.
2. what is the length of segment ( p ^ { prime } q ^ { prime } ) ? explain how you know.

Explanation:

Step1: Determine the size of the dilated triangle

When the scale factor \(k> 1\), the dilation is an enlargement. Here, the scale factor \(k = 1.5>1\). So, triangle \(P'Q'R'\) is larger than triangle \(PQR\) because a scale factor greater than \(1\) means each side length of the new triangle (the dilated one) is multiplied by a number greater than \(1\) compared to the original triangle.

Step2: Calculate the length of \(P'Q'\)

The formula for dilation of a segment length is \(L'=k\times L\), where \(L\) is the length of the original segment and \(k\) is the scale factor. Given that the length of \(PQ = 5\) and the scale factor \(k = 1.5\). Then \(P'Q'=1.5\times5\).

$$P'Q'=7.5$$

Answer:

1. Triangle \(P'Q'R'\) is larger than triangle \(PQR\). Since the scale factor \(1.5>1\), dilation with a scale factor greater than \(1\) enlarges the figure.
2. The length of segment \(P'Q'\) is \(7.5\). Using the formula \(L'=k\times L\) (where \(k = 1.5\) and \(L=PQ = 5\)), we get \(P'Q'=1.5\times5 = 7.5\).