Final answer:
Using the Pythagorean theorem, we showed that the diagonal PR of the larger square is indeed two times the length of diagonal LN of the smaller square. Additionally, comparing areas revealed that the larger square's area is 4 times the smaller square's area.
Step-by-step explanation:
Proof of Diagonal Lengths in Geometric Figures
To determine if Elisa is correct about the length of diagonal PR being two times the length of diagonal LN, we need to use the Pythagorean theorem. Suppose the side length of the smaller square (where diagonal LN is located) is d. Then, the diagonal LN will be √d² + d² = √2d². In the case of a rectangle and a square, where the rectangle is double the square in side length, and since the square's diagonal is √2 times the side length, the scale factor is also applied to the diagonal, making it 2√2 times the original side length.
The side length of Marta's original square is 4 inches and thus the diagonal is 4√2 inches. Scaling up by a factor of 2, the larger square's side length is 8 inches, and the diagonal PR is 8√2 inches. Now, if we double 4√2 inches, we get 8√2 inches, confirming that diagonal PR is indeed two times the length of diagonal LN.
Comparing the areas of the two squares involves squaring their side lengths. The area of the first square is 4 inches x 4 inches = 16 square inches. The area of the larger square is 8 inches x 8 inches = 64 square inches. The ratio of the areas is 64/16, which simplifies to 4. This shows that the area of the larger square is 4 times that of the smaller square.