Solving for the Unknown Sides of a Rectangle Using the Pythagorean Theorem
Suppose you are given the dimensions of a rectangle involving its two opposite sides and the length of its diagonal. By leveraging the Pythagorean Theorem, you can calculate the lengths of the other two sides. In this article, we'll demonstrate how to solve this problem using a step-by-step process.
Understanding the Problem
We are given that one side of the rectangle is 10 units and the diagonal is 20 units. The goal is to find the lengths of the other two sides of the rectangle.
The Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry that relates to right triangles and can be applied to rectangles, as the diagonal of a rectangle splits it into two right triangles. The theorem states that in a right 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 expressed as:
( d^2 a^2 b^2 )
where d is the length of the diagonal, and a and b are the lengths of the two sides of the rectangle.
Applying the Pythagorean Theorem to the Problem
Let's set up the equation using the given values:
[ 20^2 10^2 b^2 ]
Calculating the squares:
[ 400 100 b^2 ]
Isolating b^2:
[ b^2 400 - 100 300 ]
Now, taking the square root of both sides to find b:
[ b sqrt{300} sqrt{100 cdot 3} 10sqrt{3} ]
The value of 10sqrt{3} is approximately 17.32 units.
Therefore, the lengths of the other two sides of the rectangle are:
One side: 10 units The other side: 10sqrt{3} units (approximately 17.32 units)Conclusion
The application of the Pythagorean Theorem provided us with a straightforward method to find the unknown sides of the rectangle. The usage of these principles not only answers the immediate problem but also serves as a valuable tool in various geometric and real-world applications.