The area of the Rectangle is 240 m2
You can use the Formula for the Perimeter of a Rectangle to set up the sides in terms of one variable. Next use the Pythagorean Theorem to set up a quadratic equation and find the sides.
First for the Perimeter P = 2L + 2W, where L = Length and W = Width
P = 2L + 2W
68 = 2L + 2W
Divide both sides of the equation by 2
34 = L + W
And
34 - W = L
Next since the 26 m diagonal splits the rectangle into two equal right triangles of sides W and (34 - W) or L we can use the Pythagorean Theorem to set up a quadratic and solve for W.
(34 - W)2 - W2 = 262
1156 - 68W + W2 + W2 = 676
Combine like terms
1156 - 68W + 2W2 = 676
Subtract 676 from both sides of the equation
1156 -676 - 68W + 2W2 = 0
Combine like terms again
480 - 68W + 2W2 = 0
I will rearrange the equation here
2W2 - 68W + 480 = 0
Divide both sides of the equation by 2
W2 -34W + 240 = 0
Factor the quadratic
(w - 10)(w - 24) = 0
W = 10
W = 24
Referring back to our original set of substitutions based on the Perimeter, where L = 34 - W and L cannot be negative.
L = 34 - 10 = 24
L = 34 - 24 = 10
With
L = 24 and W = 10
OR
L = 10 and W = 24
The area of the rectangle is 240 m2
Checking with P = 68
68 = 2(24) + 2(10) = 48 + 20 = 68
68 = 2(10) + 2(24) = 20 + 48 = 68
Checking with the Pythagorean Theorem
102 + 242 = 262
100 + 576 = 676
I hope you find this useful if you have any questions please send me a message.
