A square has side lengths of 5.9 m. If this square is stretched into a rectangle with sides of 5.9 m and 6.4 m by what length has the diagonal increased?
a. 0.28 cm
c. 0.63 cm
b. 0.4 cm
d. 1.3 cm
To find the increase in the diagonal, we need to compare the diagonals of the square and the rectangle.
In a square, the diagonal is found using the Pythagorean theorem: diagonal^2 = side^2 + side^2
For a square with side length 5.9 m, the diagonal is:
diagonal_square^2 = 5.9^2 + 5.9^2
diagonal_square^2 = 2(5.9^2)
diagonal_square = √(2(5.9^2))
diagonal_square ≈ 8.352 m
In the rectangle, the diagonal can also be found using the Pythagorean theorem: diagonal^2 = side^2 + side^2
For a rectangle with sides 5.9 m and 6.4 m, the diagonal is:
diagonal_rectangle^2 = 5.9^2 + 6.4^2
diagonal_rectangle^2 = 5.9^2 + (1.5 × 5.9)^2
diagonal_rectangle^2 = 5.9^2 + (1.5^2 × 5.9^2)
diagonal_rectangle^2 = (1 + 1.5^2) × 5.9^2
diagonal_rectangle^2 = 5.225 × 5.9^2
diagonal_rectangle = √(5.225 × 5.9^2)
diagonal_rectangle ≈ 9.202 m
The increase in the diagonal is therefore:
increase = diagonal_rectangle - diagonal_square
increase = 9.202 m - 8.352 m
increase ≈ 0.85 m
However, none of the answer choices are measured in meters. Therefore, we must convert the increase to centimeters:
increase = 0.85 m × 100 cm/m
increase ≈ 85 cm
The closest answer choice to 85 cm is 0.63 cm. Therefore, the correct answer is c. 0.63 cm.