A rectangle has a diagonal of length 125. If the ratio fo the length of the short side to the length of the long side of the rectangle is 7:24, what is the length of the shorter side of the rectangle?
Recall that the length of a diagonal of a rectangle can be calculated by,
D^2 = W^2 + L^2
or simply, we'll use the Pythagorean theorem.
Substituting,
125^2 = (7x)^2 + (24x)^2
15625 = 49x^2 + 576x^2
15625 = 625x^2
x^2 = 25
x = 5 (this is the multiplier for the ratio)
Therefore, the shorter side (which is 7x) is equal to 7*5 = 35.
Hope this helps~ :3
To find the length of the shorter side of the rectangle, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the diagonal of the rectangle is the hypotenuse of a right-angled triangle, and the short side and long side of the rectangle are the other two sides.
Let's call the length of the short side "x" and the length of the long side "y". We know that the ratio of x to y is 7:24.
According to the Pythagorean theorem, we have the equation: x^2 + y^2 = 125^2.
To proceed, we can substitute x in terms of y using the given ratio. Since the ratio of x to y is 7:24, we can write x as (7/24)y.
Plugging this back into the equation, we get:
(7/24)y^2 + y^2 = 125^2.
Now, we can solve this equation for y:
(49/576)y^2 + y^2 = 15625.
Multiply through by 576 to eliminate the fractions:
49y^2 + 576y^2 = 15625 * 576.
625y^2 = 15625 * 576.
Divide both sides by 625:
y^2 = (15625 * 576) / 625.
Take the square root of both sides:
y = sqrt((15625 * 576) / 625).
Now, use a calculator to compute this expression and find the value of y:
y ≈ 60.
Therefore, the length of the longer side of the rectangle is 60.
To find the length of the shorter side, we can use the ratio: 7:24.
Since 7 corresponds to the shorter side, we can set up a proportion:
7/24 = x/60.
We can cross-multiply and solve for x:
7 * 60 = 24 * x.
420 = 24x.
Divide both sides by 24:
x ≈ 17.5.
Therefore, the length of the shorter side of the rectangle is approximately 17.5 units.