Abigail Adventuresome took a shortcut along the diagonal of a rectangular field and saved a distance equal to 1/3 the length of the longer side.Find the retio of the length of the shorter side of the rectangle to that of the longer side.
let the longer side be x and the shorter side by y
then the hypotenuse is √(x^2 + y^2)
We are told
√(x^2 + y^2) = x + x/3
√(x^2 + y^2) = 4x/3
square both sides
x^2 + y^2 = 16x^2/9
9x^2 + 9y^2 = 16x^2
9y^2 = 7y^2
take √ of both sides, using only the positive case
3y = (√7)x
y/x = √7/3
y:x = √7 : 3
let the longer side be x and the shorter side by y and diagonal = d
then the hypotenuse is √(x^2 + y^2)
d=2/3x+y
(2/3x+y)2 = (x^2 + y^2)
y/x = 5/12
that actually is wrong because i found the answer to that in the back of my book and the answer is 5/12, but i have no idea how to get to that answer
To solve this problem, we can start by assuming the shorter side of the rectangle has a length of "x" units, and the longer side has a length of "y" units. We know that Abigail saved a distance equal to 1/3 the length of the longer side by taking a shortcut along the diagonal.
Using the Pythagorean theorem, we can find the length of the 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.
In this case, the shorter side of the rectangle forms one of the legs of the right triangle, the longer side forms the other leg, and the diagonal is the hypotenuse. So, we have the equation:
(x^2 + y^2) = d^2
We also know that Abigail saved a distance equal to 1/3 the length of the longer side:
d - y = (1/3)y
Simplifying the second equation, we have:
d = (4/3)y
Substituting this value of "d" into the first equation, we have:
(x^2 + y^2) = (4/3y)^2
Simplifying further, we get:
x^2 + y^2 = 16/9y^2
Now, we need to find the ratio of the length of the shorter side to the longer side. To do this, we can divide both sides of the equation by y^2:
(x^2/y^2) + 1 = 16/9
Dividing through by 1, we have:
x^2/y^2 = 16/9 - 1 = 7/9
Taking the square root of both sides, we get:
x/y = √(7/9)
Therefore, the ratio of the length of the shorter side of the rectangle to the longer side is √(7/9), or approximately 0.8819.