A diagonal of a rectangle has length 41, and the perimeter is 98. Find the area of the rectangle.

Mellie
Dec 30, 2015

#3**+10 **

Thanks guys, Hi Mellie

A diagonal of a rectangle has length 41, and the perimeter is 98. Find the area of the rectangle.

This is a really neat completing the squares question !

2(L+B)=98

so

L+B=49

\(L^2+B^2=41^2\\ \mbox{complete the square}\\ L^2+B^2+2BL=41^2+2BL\\ (L+B)^2=41^2+2BL\\ sub\;\;L+B=49\\ 49^2=41^2+2BL\\ 49^2-41^2=2BL\\ BL=\frac{49^2-41^2}{2}\\ AREA=\frac{49^2-41^2}{2}\\ AREA=360 \;units^2\)

Melody
Dec 31, 2015

#1**+10 **

Let length=L

Let width =W

L^2 + W^2=41^2

2[L + W] =98

L = 40 and W = 9

Area= L X W=40 X 9=360

Guest Dec 30, 2015

#2**+10 **

Let length=L

Let width =W

L^2 + W^2=41^2

2[L + W] =98

2L + 2W=98

2L=98 - 2W

2L=2[49 - W] divide both sides by 2

L=49 - W Substitute

[49 - W]^2 + W^2=41^2=1,681

Solve for W:

(49-W)^2+W^2 = 1681

Expand out terms of the left hand side:

2 W^2-98 W+2401 = 1681

Divide both sides by 2:

W^2-49 W+2401/2 = 1681/2

Subtract 2401/2 from both sides:

W^2-49 W = -360

Add 2401/4 to both sides:

W^2-49 W+2401/4 = 961/4

Write the left hand side as a square:

(W-49/2)^2 = 961/4

Take the square root of both sides:

W-49/2 = 31/2 or W-49/2 = -31/2

Add 49/2 to both sides:

W = 40 or W-49/2 = -31/2

Add 49/2 to both sides:

**Answer: | W = 40 or W = 9**

Guest Dec 31, 2015

#3**+10 **

Best Answer

Thanks guys, Hi Mellie

A diagonal of a rectangle has length 41, and the perimeter is 98. Find the area of the rectangle.

This is a really neat completing the squares question !

2(L+B)=98

so

L+B=49

\(L^2+B^2=41^2\\ \mbox{complete the square}\\ L^2+B^2+2BL=41^2+2BL\\ (L+B)^2=41^2+2BL\\ sub\;\;L+B=49\\ 49^2=41^2+2BL\\ 49^2-41^2=2BL\\ BL=\frac{49^2-41^2}{2}\\ AREA=\frac{49^2-41^2}{2}\\ AREA=360 \;units^2\)

Melody
Dec 31, 2015

#4**+10 **

2(L + W) = 98

L + W = 49

L = 49 - W

L^2 + W^2 = 41^2

(49 - W)^2 + W^2 = 1681

2W^2 - 98W + 2401 = 1681

2W^2 - 98W + 720 = 0

W^2 - 49W + 360 = 0

(W - 40) (W - 9) = 0

W = 40 and L = 9 or W = 9 and L = 40 [ depending upon your orientation ]

And the area = L * W = 40 * 9 = 360 units^2

CPhill
Jan 1, 2016

#5**0 **

Let x and y be the dimensions of the rectangle. Then from the given information \(\sqrt{x^2 + y^2} = 41\) and , so x + y = 49.

Squaring the equation \(\sqrt{x^2 + y^2} = 41\), we get \(x^2 + y^2 = 1681.\)

Squaring the equation x + y= 49, we get \(x^2 + 2xy + y^2 = 2401\)

Subtracting these equations, we get 2xy = 720, so the area of the rectangle is 360 degrees.

Guest Apr 22, 2016