how to find the dimension if the diagonal of the rectangle is 8 meters longer than its shorter side and the area of the rectangle is 60 square meters?
If the dimensions are x and y, then we have
√(x^2+y^2) = x+8
xy = 60
You can grind out the solution, but I think you can recognize which factors of 60 form a convenient right triangle that works, no?
S * L = 60
S² + L² = (S + 8)²
S² + (60 / S)² = (S + 8)²
S² + (3600 / S²) = (S + 8)²
S⁴ + 3600 = S⁴ + 16 S³ + 64 S²
S³ + 4 S² - 225 = 0
S = 5
Steve is referring to the 5-12-13 Pythagorean triple
To find the dimensions of the rectangle, we can set up a system of equations based on the given information.
Let's assume that the shorter side of the rectangle is "x" meters.
According to the given information, the diagonal of the rectangle is 8 meters longer than the shorter side. So, the length of the diagonal is "x + 8" meters.
We can use the Pythagorean theorem to relate the dimensions of the rectangle to its diagonal. The Pythagorean theorem 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 (length and width).
So, in our case, we have the following equation:
(x^2) + (x + 8)^2 = (diagonal^2)
To find the area of the rectangle, we can use the formula: Area = length x width.
Since the length of the rectangle is x meters, and the width of the rectangle is x + 8 meters based on the given information, we can set up another equation:
Area = x(x + 8)
Now, we have a system of two equations:
(x^2) + (x + 8)^2 = (diagonal^2)
x(x + 8) = 60
Solving this system of equations will give us the dimensions of the rectangle.
Let's solve the second equation for x:
x(x + 8) = 60
x^2 + 8x - 60 = 0
Now we can either factor this equation or use the quadratic formula to find the solution. Factoring, we have:
(x + 12)(x - 5) = 0
So, x can be either -12 or 5. Since we are working with dimensions, which cannot be negative, we discard the -12.
Therefore, x = 5 meters.
Now, we can substitute the value of x into the first equation to find the value of the diagonal:
(x^2) + (x + 8)^2 = (diagonal^2)
(5^2) + (5 + 8)^2 = (diagonal^2)
25 + 169 = (diagonal^2)
194 = (diagonal^2)
diagonal ≈ 13.928 meters
Hence, the dimensions of the rectangle are approximately 5 meters by 13.928 meters.