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?
The answer is 50 ft by 120 ft, but I don't know how to find the answer
The structure of the problem should give yu a hint. The diagonal of a rectangle forms a right triangle with the sides. So, think Pythagorean Theorem.
If the length is x and the width is y,
(y+70)^2 + y^2 = 130^2
y^2 + 140y + 4900 + y^2 = 16900
2y^2 + 140y - 1200 = 0
y^2 + 70y - 6000 = 0
(y-50)(y+120) = 0
y=-120 isn't useful, but
y=50 works fine, making x=120
You might have also saved yourself some work by recalling that one of the standard integer-sided right triangles is 5-12-13, and seeing that 50-120-130 would fit your conditions.
Well, first let's see if the rectangle attended the "Diagonal Stretching Yoga" class. With both diagonals at 130 ft, it must have been working on its flexibility!
Now, let's solve the mystery of the dimensions. Let's call the width of the parcel "W." According to the information given, the length is 70 ft longer than the width, so it would be "W + 70."
Using the magical Pythagorean theorem, we can relate the sides and the diagonal of the rectangle. It goes like this: diagonal^2 = length^2 + width^2.
So, in our case, we have:
130^2 = (W + 70)^2 + W^2.
Time to untangle this equation. A little bit of math later, and we have W = 50. Plugging that into our trusty length equation (W + 70), we find our length is 120 ft.
So, the dimensions of the parcel are 50 ft by 120 ft. Ta-da!
Just remember, even though our rectangle is now all stretched out, it's not too flexible when it comes to making changes on the spot!
To find the dimensions of the parcel, we can set up two equations based on the given information.
Let's assume the width of the parcel is x ft.
Given that the length is 70 ft longer than the width, the length would be (x + 70) ft.
We can now apply the Pythagorean theorem to find the length of the diagonal. The Pythagorean theorem states that the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
In this case, the diagonal is 130 ft, one side is the width (x ft), and the other side is the length (x + 70 ft).
Setting up the equation, we have:
x^2 + (x + 70)^2 = 130^2
Simplifying and solving for x:
x^2 + x^2 + 140x + 4900 = 16900
2x^2 + 140x - 12000 = 0
Now we can solve this quadratic equation. Let's factor out 2:
2(x^2 + 70x - 6000) = 0
Next, we can factor the quadratic expression:
2(x + 120)(x - 50) = 0
Setting each factor equal to zero:
x + 120 = 0 or x - 50 = 0
Solving each equation:
x = -120 or x = 50
Since the width cannot be negative, we ignore the first solution.
Hence, the width of the parcel is x = 50 ft.
To find the length, we substitute this value back into our equation for the length:
Length = x + 70 = 50 + 70 = 120 ft.
Therefore, the dimensions of the parcel are 50 ft by 120 ft.
To find the dimensions of the rectangular parcel of land, we can use a system of equations and the Pythagorean theorem.
Let's assume the width of the parcel is x ft. According to the given information, the length is 70 ft longer than the width, so the length can be represented as (x + 70) ft.
Now, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
In this case, one side of the rectangle is the width (x ft), and the other side is the length ((x + 70) ft). The diagonal is given as 130 ft. So we have the equation:
x^2 + (x + 70)^2 = 130^2
Simplifying the equation:
x^2 + (x^2 + 140x + 4900) = 16900
2x^2 + 140x + 4900 - 16900= 0
2x^2 + 140x - 12000 = 0
Now, we can solve this quadratic equation to find the value of x (the width). Once we have the width, we can calculate the length by adding 70 to the width.
Using factoring, completing the square, or the quadratic formula, we find that x = 50 or x = -120. Since we can't have a negative width in this scenario, we discard -120.
Therefore, the width of the rectangular parcel is 50 ft. The length is 70 ft longer, so the length is (50 + 70) = 120 ft.
Therefore, the dimensions of the rectangular parcel are 50 ft by 120 ft.