The length and width of a rectangle are in the ratio of 5:12. If the rectangle has an area of 240 square centimeters, what is the length, in centimeters, of its diagonal?

Length / Width = 5 / 12.

Width = X CM, Length = 5/12x cm.
A = X(5/12X) = 240 cm^2.
5/12(X^2) = 240
Multiply both sides by 12:
X^2 = 576
X = 24 cm = Width
5/12X = 5/12(24) = 10 cm = Length

D^2 = W^2 + L^2 = 100 + 576 = 676

D = sqrt(675) = 26 cm = Diagonal.

To find the length and width of the rectangle, we can set up a system of equations. Let's denote the length as 5x and the width as 12x, where x is a common factor.

The area of a rectangle is given by the formula:
Area = Length * Width

We are given that the area is 240 square centimeters:
240 = (5x) * (12x)

To solve for x, let's divide both sides of the equation by 60:
240/60 = (5x * 12x) / 60
4 = x^2

Taking the square root of both sides, we get:
√4 = √x^2
2 = x

Now that we know x = 2, we can find the length and the width of the rectangle:
Length = 5x = 5 * 2 = 10 centimeters
Width = 12x = 12 * 2 = 24 centimeters

To find the diagonal of the rectangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the length and width of the rectangle form the two sides of a right triangle, and the diagonal is the hypotenuse.

Using the Pythagorean theorem, we have:
Diagonal^2 = Length^2 + Width^2

Substituting the values, we get:
Diagonal^2 = 10^2 + 24^2
Diagonal^2 = 100 + 576
Diagonal^2 = 676

Taking the square root of both sides, we find:
Diagonal = √676
Diagonal = 26 centimeters

Therefore, the length of the diagonal of the rectangle is 26 centimeters.

To find the length and width of the rectangle, we can use the ratio given. Let's assume the length is 5x and the width is 12x, where x is a common ratio.

Now, we can use the formula for the area of a rectangle: Area = Length × Width. In this case, we know the area is 240 square centimeters. So we can substitute the values into the equation:

240 = (5x) × (12x)

Now, let's solve for x by dividing both sides of the equation by 60:

240/60 = (5x) × (12x) / 60
4 = x²

Taking the square root of both sides, we find:

√4 = x
x = 2

Now, we know that x = 2, so we can substitute it back into the equations we assumed earlier:

Length = 5x = 5(2) = 10 cm
Width = 12x = 12(2) = 24 cm

To find the length of the diagonal, we can use the Pythagorean theorem. In a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides. In this case, the length and width of the rectangle form the two sides of the triangle.

So, the length of the diagonal (d) can be found using the equation:

d² = Length² + Width²

Substituting the values we found earlier:

d² = 10² + 24²
d² = 100 + 576
d² = 676

Now, taking the square root of both sides, we find:

d = √676
d = 26 cm

Therefore, the length of the diagonal of the rectangle is 26 centimeters.