Finding the hypotenuse of a right-angled triangle might sound intimidating, but it's actually quite straightforward! This guide will walk you through the process, explaining the concept and providing examples to solidify your understanding.
What is the Hypotenuse?
The hypotenuse is the longest side of a right-angled triangle. It's always the side opposite the right angle (the 90-degree angle). Understanding this is the first step to calculating its length.
How to Calculate the Hypotenuse: The Pythagorean Theorem
The key to finding the hypotenuse is the Pythagorean Theorem. This fundamental theorem of geometry states:
a² + b² = c²
Where:
- a and b represent the lengths of the two shorter sides (legs) of the right-angled triangle.
- c represents the length of the hypotenuse.
This equation tells us that the sum of the squares of the two shorter sides equals the square of the hypotenuse.
Steps to Calculate the Hypotenuse:
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Identify the legs (a and b): Make sure you know the lengths of the two shorter sides of your right-angled triangle.
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Square the legs: Square each of the lengths you identified in step 1 (a² and b²).
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Add the squares: Add the two squared values together (a² + b²).
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Find the square root: Take the square root of the sum you calculated in step 3. This will give you the length of the hypotenuse (c).
Examples:
Let's work through a couple of examples:
Example 1:
A right-angled triangle has legs of length 3 and 4. Find the hypotenuse.
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a = 3, b = 4
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a² = 9, b² = 16
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a² + b² = 9 + 16 = 25
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√25 = 5
Therefore, the hypotenuse has a length of 5.
Example 2:
A right-angled triangle has legs of length 5 and 12. Find the hypotenuse.
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a = 5, b = 12
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a² = 25, b² = 144
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a² + b² = 25 + 144 = 169
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√169 = 13
Therefore, the hypotenuse has a length of 13.
Beyond the Basics: Finding a Leg
The Pythagorean Theorem isn't just for finding the hypotenuse. If you know the hypotenuse and one leg, you can easily solve for the other leg! Simply rearrange the formula:
a² = c² - b² or b² = c² - a²
Conclusion
Mastering the Pythagorean Theorem opens up a world of possibilities in geometry and related fields. Remember the simple steps, practice with different examples, and you'll become a hypotenuse-finding pro in no time!
