Answer: In a rectangle, the diagonals are congruent and bisect each other, forming 4 right triangles. We can use the Pythagorean theorem to solve for the length of the diagonals:
Let d be the length of the diagonals.
Then, by the Pythagorean theorem, we have:
(5x - 31)^2 + (d/2)^2 = (2x + 11)^2 + (d/2)^2
Expanding and simplifying, we get:
25x^2 - 310x + 961 + d^2/4 = 4x^2 + 44x + 121 + d^2/4
Subtracting 4x^2 + 44x + 121 + d^2/4 from both sides, we get:
21x^2 - 354x + 840 = 0
Dividing both sides by 21, we get:
x^2 - 17x + 40 = 0
Factoring, we get:
(x - 5)(x - 12) = 0
So, x = 5 or x = 12.
If x = 5, then QS = 5(5) - 31 = -6, which is not a valid length for a side of a rectangle.
If x = 12, then QS = 5(12) - 31 = 29 and RT = 2(12) + 11 = 35.
So, the length of the diagonals is:
d = sqrt(QS^2 + RT^2) = sqrt(29^2 + 35^2) ≈ 46.8
Therefore, the length of the diagonals of QRST is approximately 46.8 units.
Step-by-step explanation: