The area of a rug is 108 square feet and the length of its diagonal is 15 feet. Find the length and width of the rug.I don't really need an answer, but the process I would use to get the answer would be helpful. I'm assuming you have to wet up two equations to solve it. If anyone could help, it would be well appreciated. There's a picture in the book of a rectangular rug with the diagonal going through it.
TRICKY ANSWER trio 15, 12, 9 gives diagonal 15, legs 12 and 9 and area 12*9 = 108.
Let x denote the length of the rug and y denote the width of the rug. Since the length, width, and diagonal of the rug form a right triangle, we have that, by the Pythagorean Theorem: x^2 + y^2 = 15^2 == x^2 + y^2 = 225. . . . . . . . . . . .(1) Then, since the area of the rug is 108 ft^2, we have: xy = 108. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(2) By solving (2) for y, we have: y = 108/x. Substituting this into (1) gives: x^2 + (108/x)^2 = 225 == x^2 + 11664/x^2 = 225 == x^4 + 11664 = 225x^2, by multiplying both sides by x^2 == x^4 - 225x^2 + 11664 = 0 == (x + 12)(x - 12)(x + 9)(x - 9) = 0, by factoring == x = ±9 and x = ±12, by the zero-product property. Since the length of the rug must be positive: x = 9 and x = 12. From this, you can obtain that y = 12 and y = 9. Regardless, the dimensions of the rug are 9 ft by 12 ft. I hope this helps!
