Finding the Perimeter of a Rectangle with Given Dimensions and Diagonal

Understanding the properties and measurements of geometric shapes is an essential skill in geometry. One of the common problems involving rectangles is determining the perimeter when given specific dimensions and the length of a diagonal. In this article, we will walk through a detailed example to explain how to find the perimeter of a rectangle, given its length and the length of its diagonal.

Introduction to Perimeter and Diagonal in Rectangles

A rectangle is a quadrilateral with four right angles. The perimeter of a rectangle is the total length of its boundaries. The diagonal of a rectangle is a line segment that connects two opposite corners. Knowing the measurements of the length, width, and diagonal of a rectangle can help in solving various geometric problems.

Example Problem: Finding the Perimeter Given Length and Diagonal

We will use the following example to illustrate the process of finding the perimeter of a rectangle when given its length (20 cm) and the length of its diagonal (21 cm).

Step-by-Step Solution

Identify the given information: - Length of the rectangle, ( l 20 ) cm - Length of the diagonal, ( d 21 ) cm

Use the Pythagorean theorem to find the width or height of the rectangle. The Pythagorean theorem states that for any right-angled triangle, the square of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides (length and width).

Formula: ( d^2 l^2 w^2 )

Substitute the known values into the formula and solve for the width (w): [ 21^2 20^2 w^2 ] [ 441 400 w^2 ] [ w^2 441 - 400 ] [ w^2 41 ] [ w sqrt{41} ] [ w ≈ 6.40 ] cm

Calculate the perimeter of the rectangle using the formula for the perimeter of a rectangle, ( P 2(l w) ), where ( l ) is the length and ( w ) is the width.

Formula: ( P 2(l w) )

Substitute the values: [ P 2(20 6.40) ] [ P 2(26.4) ] [ P 52.8 ] cm

State the final answer:

The perimeter of the rectangle is 52.8 cm.

Key Concepts to Remember

Pythagorean Theorem: This theorem is crucial for solving problems involving right triangles and is the basis for finding the width of the rectangle in this example. Perimeter Formula: The perimeter of a rectangle can be calculated using the formula ( P 2(l w) ), where ( l ) is the length and ( w ) is the width.

Conclusion

Determining the perimeter of a rectangle given its length and diagonal involves using the Pythagorean theorem to find the width and then applying the perimeter formula. This method not only helps in solving specific geometric problems but also enhances understanding of the relationships between different dimensions of a rectangle.