Finding the Length of a Rectangle: A Step-by-Step Guide Using the Pythagorean Theorem

In this tutorial, we will explore how to find the length of a rectangle when its breadth and diagonal are given. The Pythagorean theorem is a fundamental concept in geometry that allows us to solve such problems easily. Let's delve into the details with a practical example.

The Pythagorean Theorem and Its Application to Rectangles

The Pythagorean theorem is a statement that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is represented by the equation:

c2 a2 b2, where:

c is the hypotenuse a and b are the other two sides of the triangle.

In a rectangle, the diagonal forms a right triangle with the length and breadth. Therefore, we can use the Pythagorean theorem to find the length of the rectangle.

Example: Given Breadth and Diagonal

Suppose the breadth of a rectangle is 10 cm and its diagonal is 26 cm. Our goal is to find the length of the rectangle.

Step-by-Step Solution

Identify the Known Values: Breadth (b) 10 cm Diagonal (d) 26 cm Apply the Pythagorean Theorem:

According to the theorem:

d^2 l^2 b^2

Substituting the given values:

26^2 l^2 10^2
Calculate the Squares:

26^2 676

10^2 100

Thus:

676 l^2 100
Isolate the Unknown Variable (Length l):

Subtract 100 from both sides:

676 - 100 l^2

576 l^2

Solve for l:

Take the square root of both sides:

l sqrt{576} 24,text{cm}

Therefore, the length of the rectangle is 24 cm.

Alternative Calculation

We can also solve for the length using the formula:

l sqrt{d^2 - b^2}

Substituting the values:

l sqrt{26^2 - 10^2} sqrt{676 - 100} sqrt{576} 24,text{cm}

Thus, the length of the rectangle is again 24 cm.

Concluding the Calculation

Now, we can determine the area of the rectangle:

text{Area} l times b 24 times 10 240,text{cm}^2

The area of the rectangle is 240 cm2.

Key Takeaways

Understanding the Pythagorean theorem is crucial when dealing with right triangles, especially rectangles. The formula (d^2 l^2 b^2) helps in finding the unknown side of a right triangle. The length of the rectangle can be calculated using the given diagonal and breadth. The area of the rectangle can be determined by multiplying the length and breadth.

In conclusion, the length of the rectangle is 24 cm, and the area is 240 cm2. This method can be applied to any rectangle where the diagonal and one side are known.