As you may remember from your eighth-grade math class, a is the longest side of a . Finding the hypotenuse is important for many reasons, such as when calculating the distance between two points or determining the length of a ladder needed to reach a certain height. Luckily, finding the hypotenuse is a straightforward process that anyone can master with a little practice.

The first thing you need to know is the Pythagorean Theorem, named after the ancient Greek mathematician Pythagoras. The theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In math notation, it looks like this:

a² + b² = c²

where a and b are the two shorter sides and c is the hypotenuse.

Now that we know the theorem, let’s apply it to find the hypotenuse. Suppose we have a right triangle with sides a = 3 and b = 4. To find the hypotenuse, we plug a and b into the formula and solve for c:

3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25
c = 5

Therefore, the hypotenuse of a right triangle with sides 3 and 4 is 5. Easy enough, right?

But what if we don’t know the lengths of the shorter sides? In that case, we can use the inverse of the Pythagorean Theorem, known as the square root of the sum of squares. The formula looks like this:

c = √(a² + b²)

Let’s apply this formula to another example. Suppose we have a right triangle with sides a = 5 and b = 12. To find the hypotenuse, we plug a and b into the formula and solve for c:

c = √(5² + 12²)
c = √(25 + 144)
c = √169
c = 13

Therefore, the hypotenuse of a right triangle with sides 5 and 12 is 13. Again, not too difficult.

But what about non-right triangles? Can we find the hypotenuse in those cases? The answer is no, because the hypotenuse is defined as the longest side of a right triangle, and non-right triangles don’t have a hypotenuse. However, we can still use the Pythagorean Theorem to find the length of the longest side in any triangle, as long as we know the lengths of the other two sides.

Suppose we have a triangle with sides a = 5, b = 7, and c = 9. Is this a right triangle? To find out, we can check if the sum of the squares of the two shorter sides is equal to the square of the longest side:

5² + 7² = 25 + 49 = 74
9² = 81

Since 74 ≠ 81, the triangle is not a right triangle and therefore doesn’t have a hypotenuse. However, we can still use the Pythagorean Theorem to find the length of the longest side. By rearranging the formula, we get:

c² = a² + b² – 2ab cos(C)

where C is the angle opposite the longest side. In our example, we can use the Law of Cosines to find the angle C:

cos(C) = (a² + b² – c²) / 2ab
cos(C) = (5² + 7² – 9²) / 2(5)(7)
cos(C) = 0.125
C ≈ 82.6°

Now we can plug in the values and solve for c:

c² = 5² + 7² – 2(5)(7) cos(82.6°)
c² = 25 + 49 – 70(0.125)
c ≈ 5.4

Therefore, the longest side of the triangle is approximately 5.4. Note that in this case, the longest side is not called the hypotenuse, because it does not belong to a right triangle.

In conclusion, finding the hypotenuse is simply a matter of using the Pythagorean Theorem. Whether you need to the distance between two points, find the length of a ladder, or solve a geometry problem, knowing how to find the hypotenuse can make your life easier. So go ahead and practice this skill until you can do it in your sleep!

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