Calculating Rectangle Area from Length and Diagonal

The problem asks us to find the area of a rectangle when its length and the length of its diagonals are known. We are given that the length of the rectangle is 24 cm and the length of each diagonal is 25 cm.

To find the area of a rectangle, we need both its length and its width. The formula for the area of a rectangle is:

\(\text{Area} = \text{Length} \times \text{Width}\)

We are given the length (\(L = 24\) cm) but not the width (\(W\)). However, we are given the diagonal length (\(D = 25\) cm).

Using the Pythagorean Theorem to Find the Width

In a rectangle, the adjacent sides and a diagonal form a right-angled triangle. The length and the width are the two legs of this right-angled triangle, and the diagonal is the hypotenuse.

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. For our rectangle, this means:

\(\text{Length}^2 + \text{Width}^2 = \text{Diagonal}^2\)

Substituting the given values:

\(24^2 + W^2 = 25^2\)

Now, let's calculate the squares of the given numbers:

• \(24^2 = 24 \times 24 = 576\)
• \(25^2 = 25 \times 25 = 625\)

So the equation becomes:

\(576 + W^2 = 625\)

To find \(W^2\), we subtract 576 from both sides:

\(W^2 = 625 - 576\)

\(W^2 = 49\)

To find the width \(W\), we take the square root of 49:

\(W = \sqrt{49}\)

\(W = 7\) cm

Since the width must be a positive value, the width of the rectangle is 7 cm.

Calculating the Area of the Rectangle

Now that we have both the length and the width, we can calculate the area:

\(\text{Area} = \text{Length} \times \text{Width}\)

\(\text{Area} = 24 \text{ cm} \times 7 \text{ cm}\)

\(\text{Area} = 168 \text{ cm}^2\)

The area of the rectangle is 168 cm².

Comparing this result with the given options, we find that it matches option 4.

Revision Table: Key Concepts

Additional Information: Pythagorean Triples

The numbers 7, 24, and 25 form a special set called a Pythagorean triple. A Pythagorean triple consists of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\). In this problem, 7, 24, and 25 satisfy this condition (\(7^2 + 24^2 = 49 + 576 = 625 = 25^2\)). Recognizing common Pythagorean triples (like 3-4-5, 5-12-13, 7-24-25, 8-15-17) can sometimes help solve geometry problems more quickly.

Understanding how the diagonal, length, and width of a rectangle relate through the Pythagorean theorem is fundamental for solving such geometry problems involving rectangles.