The area of the square is 2,500; its sidelength is the square root of this:

This means that the right triangle in the diagram has legs  and 40 and hypotenuse 50, so by the Pythagorean Theorem:

The perimeter of  Quadrilateral  is the sum of the lengths of .

 is a side of the square and has length 2. 

 is the midpoint of , so  is half a side of the square; its length is 1.

The other two sides are hypotenuses of right triangles and can be found using the Pythagorean Theorem.

 and  are the midpoints of  and  , respectively,both of which are segments with length 2; therefore, 

.

 is therefore the hypotenuse of a triangle with legs 1 and 1, and its length is 

.

Similarly,  is the hypotenuse of a triangle with legs 1 and 2, and its length is 

.

Add these:

This rounds to 6.7.

To determine the other side length, we will need to use the Pythagorean Theorem.

Substitute the hypotenuse and the known side length as either  or .

Subtract 36 from both sides and reduce.

Square root both sides and reduce.

The length and width of the triangle are now known.

Write the formula for the area of a triangle.

Substitute the dimensions.

The answer is .

To find the other side length, we will need to first use the Pythagorean Theorem.

Substitute the side and hypotenuse.

Solve for the missing side.

Write the formula for the area of a triangle.

Substitute the sides.

The answer is:  

Write the formula for the Pythagorean Theorem.

Substitute the values into the equation.

Subtract 25 from both sides.

Square root both sides.

The answer is:  

Write the Pythagorean Theorem to find the hypotenuse.

Substitute the dimensions.

Square root both sides.

The answer is:  

Start by drawing out what the car did.

You'll notice that a right triangle will be created as shown by the figure above. Thus, the shortest distance between the car and City A is also the hypotenuse of the triangle. Use the Pythagorean Theorem to find the distance between the car and City A.

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