How to Find Hypotenuse Using Pythagoras’ Theorem

Finding Hypotenuse Sides using Pythagoras Theorem

Pythagoras’ Theorem is a fundamental concept in mathematics that allows you to find the length of the hypotenuse in a right-angled triangle when you know the lengths of the other two sides. This powerful theorem has numerous applications in various fields, from architecture and engineering to navigation and astronomy. In this article, we’ll explore the basics of Pythagoras’ Theorem and provide you with a step-by-step guide on how to find the hypotenuse using this essential mathematical tool.

Understanding Pythagoras’ Theorem

Before we dive into the process of finding the hypotenuse, let’s first understand what Pythagoras’ Theorem is and how it works.

The Theorem Statement

Pythagoras’ Theorem states 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 (called the base and perpendicular).

Mathematically, this can be expressed as:

\(a^2 + b^2 = c^2\)

Where:

  • \(a\) and \(b\) are the lengths of the base and perpendicular
  • \(c\) is the length of the hypotenuse

The Importance of the Right Angle

It’s crucial to note that Pythagoras’ Theorem only applies to right-angled triangles. A right-angled triangle is a triangle that has one 90-degree angle (right angle). The side opposite the right angle is always the longest side and is called the hypotenuse.

Step-by-Step Guide to Finding the Hypotenuse

Now that we understand the basics of Pythagoras’ Theorem, let’s explore how to use it to find the length of the hypotenuse.

Step 1: Identify the Right-Angled Triangle

The first step in finding the hypotenuse is to identify the right-angled triangle in your problem. Look for a triangle with one 90-degree angle, and label the sides accordingly:

  • The side opposite the right angle is the hypotenuse (\(c\))
  • The other two sides are the base (\(a\)) and perpendicular (\(b\))

Step 2: Determine the Known Side Lengths

Next, determine the lengths of the base and perpendicular. These lengths will be given to you in the problem or can be calculated using other given information.

Step 3: Set Up the Equation

Once you have the lengths of the base and perpendicular, set up the equation using Pythagoras’ Theorem:

\(a^2 + b^2 = c^2\)

Substitute the known values for \(a\) and \(b\) into the equation.

Step 4: Solve for the Hypotenuse

To solve for the hypotenuse (\(c\)), isolate \(c^2\) on one side of the equation by adding the squares of the base and perpendicular:

\(c^2 = a^2 + b^2\)

Then, take the square root of both sides to find the length of the hypotenuse:

\(c = \sqrt{a^2 + b^2}\)

Step 5: Simplify and Check Your Answer

Simplify the expression under the square root if possible, and calculate the final value of the hypotenuse. Always double-check your answer to ensure that it makes sense in the context of the problem.

Example Problem

Let’s apply the steps we’ve learned to solve an example problem.

Problem: In a right-angled triangle, the base has a length of 3 units, and the perpendicular has a length of 4 units. Find the length of the hypotenuse.

Solution:

Step 1: Identify the right-angled triangle and label the sides.

  • Base (\(a\)) = 3 units
  • Perpendicular (\(b\)) = 4 units
  • Hypotenuse (\(c\)) = unknown

Step 2: The known side lengths are already given.

Step 3: Set up the equation using Pythagoras’ Theorem.
\(3^2 + 4^2 = c^2\)

Step 4: Solve for the hypotenuse.
\(c^2 = 3^2 + 4^2\)
\(c^2 = 9 + 16\)
\(c^2 = 25\)
\(c = \sqrt{25}\)
\(c = 5\)

Step 5: The length of the hypotenuse is 5 units.

Common Mistakes to Avoid

When using Pythagoras’ Theorem to find the hypotenuse, there are some common mistakes to watch out for:

  1. Confusing the sides: Make sure to identify the hypotenuse, base, and perpendicular correctly. Remember that the hypotenuse is always the side opposite the right angle.
  2. Forgetting to square the side lengths: When setting up the equation, be sure to square the lengths of the base and perpendicular before adding them together.
  3. Misapplying the theorem to non-right-angled triangles: Pythagoras’ Theorem only works for right-angled triangles. Please don’t attempt to use it on other types of triangles.
  4. Miscalculating the square root: When solving for the hypotenuse, ensure that you correctly calculate the square root of the sum of the squares of the base and perpendicular.

By avoiding these common mistakes, you’ll be well on your way to mastering the use of Pythagoras’ Theorem for finding the hypotenuse.

Conclusion

Pythagoras’ Theorem is a powerful tool for finding the length of the hypotenuse in a right-angled triangle. By understanding the theorem and following the step-by-step guide provided in this article, you’ll be able to confidently tackle problems involving the hypotenuse and unlock the full potential of this essential mathematical concept.

Remember to practice applying Pythagoras’ Theorem to a variety of problems to reinforce your understanding and develop your problem-solving skills. With dedication and persistence, you’ll soon find yourself mastering the art of finding the hypotenuse and marveling at the elegance and simplicity of Pythagoras’ Theorem.

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