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

In order to find the distance from the top of the tree to the end of the shadow, draw a right triangle with the height(tree) labeled as 8 and base(shadow) labeled as 6: 

From this diagram, you can see that the distance being asked for is the hypotenuse. From here, you can either use the Pythagorean Theorem:

or you can notice that this is simililar to a 3-4-5 triangle. Since the lengths are just increased by a factor of 2, the hypotenuse that is normally 5 would be 10. 

Assigning the length of ED the value of x, the value of AE will be 3x. That makes the entire side AD equal to 4x. Since the figure is a square, all four sides will be equal to 4x. Also, since the figure is a square, then angle A of triangle ABE is a right angle. That gives triangle ABE sides of 3x, 4x and 10. Using the Pythagorean theorem:

(3x)² + (4x)² = 10²

9x² + 16x² = 100

25x² = 100

x² = 4

x = 2

With x = 2, each side of the square is 4x, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.

Using the Pythagorean Theorem, we can find the length of the hypotenuse:

.

Therefore the hypotenuse has length 5.

The area of the circle is

To find the length of this hypotenuse, we need to use the Pythagorean Theorem:

, where a and b are the legs and c is the hypotenuse.

Here, c is our missing hypotenuse length, a = 4 ,and b = 14.

Plug these values in and solve for c:

Use the Pythagorean Theorem: , where a and b are the legs and c is the hypotenuse.

We know  and , so we can plug them in to solve for c: