There are two ways to solve this problem. The first is to recognize that the right triangle follows the pattern of a well-known Pythagorean triple: .

The second is to use the Pythagorean Theorem:

 , where and are the lengths of the triangle sides and is the length of the hypotenuse.

Plugging in our values, we get:

In a right triangle, the base and the height are the legs of the triangle.  To find the hypotenuse, square both legs and add the results together.  Then, find the square root of that sum.

Jeff drives a total of 10 miles north and 10 miles east.  Using the Pythagorean theorem (a²+b²=c²), the direct route from Jeff’s home to his work can be calculated.  10²+10²=c².  200=c². √200=c. √100Ÿ√2=c. 10√2=c

By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:

10² + 5² = x²

100 + 25 = x²

√125 = x, but we still need to factor the square root

√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so

5√5= x

In order to find the length of the diagonal accross a square, we must first find the lengths of the individual sides.

The area of a square is found by multiply the lengths of 2 sides of a square by itself.

So, the square root of 3,600 comes out to 60 ft.

The diagonal of a square can be found by treating it like a right triangle, and so, we can use the pythagorean theorem for a right triangle.

60² + 60² = C²

the square root of 7,200 is 84.8, which can be rounded to 85

Pythagorean Theorum

AB² + BC² = AC²

If C is 45º then A is 45º, therefore AB = BC

AB² + BC² = AC²

6² + 6² = AC²

2*6² = AC²

AC = √(2*6²) = 6√2

Distance = hours * mph

North Distance = 5 hours * 60 mph = 300 miles

East Distance = 2 hours * 50 mph = 100 miles

Use Pythagorean Theorem to determine Northeast Distance

300^{2 +} 100² =NE²

90000  + 10000 = 100000 = NE²

NE = √100000

Since the garden is square, the two sides are equal to the square root of the area, making each side 7 feet. Then, using the Pythagorean Theorem, set up the equation 7² + 7² = the length of the diagonal squared. The length of the diagonal is the square root of 98, which is closest to 10.

We look at the 3 points of interest: the lighthouse, where the pelican started, and where the pelican ended. We can see that if we connect these 3 points with lines, they form a right triangle. (From due north, flying exactly west creates a 90 degree angle.) The three sides of the triangle are 5 miles, 13 miles and an unknown distance. Using the Pythagorean Theorem we get:

A right triangle can be drawn between the airplane and its destination.

                           Destination

                      15 miles    Airplane

                                     8 miles

We can solve for the hypotenuse, x, of the triangle:

8² + 15² = x²

64 + 225 = x²

289 = x²

x = 17 miles