+45 A rectangular sheet of paper \(8\frac{1}{2}\) inches by 11 inches is folded and then taped so that one pair of opposite vertices coincide. What is the number of inches in the perimeter of the resulting pentagon? Express your answer as a decimal to the nearest tenth.
+179 Folding the sheet of paper as described brings vertex B to coincide with vertex D.
The resulting figure is a pentagon with side lengths AB, BC, CD, DE, and EA.
We know that AB=8.5 inches and AD=11 inches. Since ABCD is a rectangle, CD=AB=8.5 inches.
Segment DE is the hypotenuse of a right triangle with legs AD=11 inches and BE=2AB=4.25 inches.
By the Pythagorean Theorem, we can find DE=112+4.252≈156.25≈12.5 inches (rounded to the nearest tenth).
The final side length, EA, is the hypotenuse of another right triangle with legs AB=8.5 inches and BE=4.25 inches.
Using the Pythagorean Theorem again, we find that EA=8.52+4.252≈90.25≈9.5 inches (rounded to the nearest tenth).
Adding up all the side lengths, we find the perimeter of the pentagon is
AB + BC + CD + DE + EA = 8.5 + 8.5 + 8.5 + 12.5 + 9.5 = 47.5 inches (to the nearest tenth).
