A rectangular parcel of land is 70 ft longer than it is wide. Each diagonal between opposite corners is 130 ft. What are the dimensions of the parcel?
sides are x and x+70
x^2 + (x+70)^2 = 130^2
x^2 + x^2 + 140x + 4900 = 16900
2x^2 + 140x - 12000 = 0
x^2 + 70x - 6000 = 0
(x+120)(x-50) = 0
x = 50
sides are 50, 120
Or, more easily, you could have checked for a scaled-up 5,12,13 triangle. The diagonal of 130 was a good hint.
To find the dimensions of the rectangular parcel of land, let's first assign variables to the dimensions. Let's say the width of the parcel is 'w' feet.
According to the problem, the length of the parcel is 70 feet longer than the width. Therefore, the length would be 'w + 70' feet.
Now, let's use the Pythagorean theorem to solve for the dimensions. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (width and length). In our case, the diagonal is given as 130 feet.
Using the Pythagorean theorem, we can write the equation as:
(width)^2 + (length)^2 = (diagonal)^2
Substituting the values we know:
w^2 + (w + 70)^2 = 130^2
Expanding and simplifying the equation:
w^2 + w^2 + 140w + 4900 = 16900
Combining like terms:
2w^2 + 140w + 4900 - 16900 = 0
Simplifying further:
2w^2 + 140w - 12000 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = 140, and c = -12000:
w = (-140 ± √(140^2 - 4*2*-12000)) / (2*2)
Simplifying further:
w = (-140 ± √(19600 + 96000)) / 4
w = (-140 ± √(115600)) / 4
w = (-140 ± 340) / 4
Solving for both values of w:
w1 = (-140 + 340) / 4
w1 = 200 / 4
w1 = 50
w2 = (-140 - 340) / 4
w2 = -480 / 4
w2 = -120
Since we are dealing with dimensions, the width cannot be negative. Therefore, we discard w2 = -120.
The width of the parcel is w = 50 feet.
To find the length of the parcel, we substitute the width value into our earlier equation:
length = width + 70
length = 50 + 70
length = 120
Therefore, the dimensions of the rectangular parcel are 50 ft (width) and 120 ft (length).