How do you apply Pythagoras' Theorem to find the diagonal of a rectangle?

To find the diagonal of a rectangle, use Pythagoras' Theorem: \(d = \sqrt{l^2 + w^2}\).

Pythagoras' Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the case of a rectangle, the diagonal forms the hypotenuse of a right-angled triangle, with the rectangle's length and width as the other two sides.

Let's break it down step-by-step. Suppose you have a rectangle with a length \(l\) and a width \(w\). To find the length of the diagonal \(d\), you can imagine drawing a right-angled triangle within the rectangle, where the diagonal is the hypotenuse. According to Pythagoras' Theorem, the relationship between the sides is given by:

\[d^2 = l^2 + w^2\]

To find \(d\), you need to take the square root of both sides of the equation:

\[d = \sqrt{l^2 + w^2}\]

For example, if the length of the rectangle is 3 units and the width is 4 units, you would calculate the diagonal as follows:

\[d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]

So, the diagonal of a rectangle with a length of 3 units and a width of 4 units is 5 units. This method can be applied to any rectangle, regardless of its dimensions, making it a versatile and essential tool in geometry.