Question

Noah drew a scaled copy of Polygon P and labeled it Polygon Q. The area of the original, Polygon P, is 5 square units. How many times larger is the area of this polygon (Polygon Q)? What scale factor did Noah apply to each length in Polygon P to create Polygon Q?

Blank 1: 25
Blank 2: 5

Answer 1

The area of Polygon Q is 25 times larger than Polygon P, and Noah applied a scale factor of 5 to each length in Polygon P to create Polygon Q.

Step-by-step explanation:

Noah used a scaling factor to create Polygon Q from Polygon P. The area of a polygon is directly proportional to the square of its linear dimensions when the shape is scaled uniformly. In this case, the area of Polygon Q is 25 times larger than that of Polygon P. This relationship arises from the fact that the area scales with the square of the linear dimensions.

To find the scale factor applied to each length in Polygon P, we take the square root of the area scale factor. The square root of 25 is 5. This means that each length in Polygon Q is 5 times the corresponding length in Polygon P. The scale factor is a multiplier that determines the proportional relationship between corresponding lengths in the two polygons.

In mathematical terms, if the lengths of Polygon P are represented as L1, L2, ..., Ln, then the lengths of Polygon Q are L1 × 5, L2 × 5, ..., Ln × 5. The area of Polygon Q is calculated by squaring this scale factor: (5)^2 = 25. Therefore, the area of Polygon Q is 25 times larger than the area of Polygon P.