You want to build a garden in the shape of a right triangle. If the two arms will be 6ft and 8ft, what does the length of the hypotenuse need to be?

To find the length of a hypotenuse of a right triangle, simply use the Pythagorean Theorem.

Where a and b are the arm lengths, and c is the hypotenuse.

Plug in our knowns and solve.

Note that we could also have found c by identifying a Pythagorean Triple:

3x-4x-5x

3(2)-4(2)-5(2)

6-8-10 

By the Pythagorean Theorem, if  and  are the lengths of the shorter two sides, or legs, of a right triangle, and  is the length of the longest side, or hypotenuse, of the triangle, then 

Set :

Since we are trying to determine between which two of the consecutive integers in the set   falls, it suffices to find out between which two of their squares  falls. Each square is the product of the integer and itself, so:

,

or, substituting, 

It follows that 

.

You are visiting a friend who has right-triangular shaped pool. You are seeing who can swim around the perimeter of the pool fastest. If the long side is 20 meters, and second shortest side is 15 meters long, how long is the shortest side?

Let's begin by recalling Pythagorean Theorem

So, we know that c is our hypotenuse or longest side. 

a and b are our shorter sides. It doesn't really matter which one is which.

Let's plug in and solve!

So, our answer is 

Notice that the hypotenuse of the right triangle  is also the length of the rectangle.

Start by using Pythagorean's Theorem to find the length of .

Next, recall how to find the area of a rectangle:

By the Pythagorean Theorem, if we let be the length of the hypotenuse, or longest side, of a right triangle, and and be the lengths of the legs, the relation is

Set and , and solve for :

Square the numbers - that is, multiply them by themselves:

Subtract 81 from both sides to isolate :

To find out what integers falls between, it is necessary to find the perfect square integers that flank 319. We can see by trial and error that

,

so

The length of the second leg thus falls between 17 and 18.

You recently bough a book end whose face forms a right triangle. You want to know the length of the longest side, but you don't have a ruler. Luckily, you know that the two shorter sides are 18 inches and 24 inches. Find the length of the last side.

The problem describes a right triangle with two known sides. Finding the last side sounds like a job for Pythagorean Theorem. If you look carefully, you might see another way.

So, recall the classic:

We know a and b, and we need to find c.

So our answer is 30.

The alternate way to solve this is to see that we have a 3/4/5 Pythagorean Triple.

This means that our side lengths follow the ratio 3:4:5

We can see this by dividing our two given sides by 6.

So our scale factor is 6. This means we can get our answer by multiplying 5 and 6

So we get the same thing. Keep an eye out for Pythagorean Triples when working with right triangles in order to solve problems faster!

Start by drawing out the trapezoid in question.

Notice that triangle  is a right triangle. Thus, we can use the Pythagorean theorem to find the length of segment .

Since we know that  is also , then that means the length of segment  must also be .

Now, we can use the Pythagorean Theorem again to find the length of .

Now we can find the perimeter of the entire isosceles trapezoid.

Make sure to round to two places after the decimal.

Notice that the diagonal of the rectangle is also the diameter of the circle. We can use the Pythagorean Theorem to find the length of the diagonal.

From this, we can find the radius of the circle.

Recall how to find the area of a circle.

Plug in the given radius.

We need to use the Pythagorean Theorem to find the missing side;

The theorem says:

"For any right triangle, with legs  and  and hypotenuse , the formula  can be used to find any missing side of this triangle."

So, we are given  and  in the question...

We will plug them into the theorem:

Simplify:

Simplify:

Add:

To find , we must take the square root of both sides:

So..

Reduce:

Simplify:

For this problem, you just need to remember your handy Pythagorean theorem. Remember that it is defined as:

where  and  are the legs of the triangle, and  is the hypotenuse. Remember, however, that this only works for right triangles. Thus, based on your data, you know:

or

Subtracting 484 from each side of the equation, you get:

Using your calculator to calculate the square root, you get:

Rounding, this is , so the triangle's other leg measures .