Kathy's path traces the outline of a right triangle with legs of 6 and 8. By using the Pythagorean Theorem

  \(\displaystyle \dpi{100} \small 6^{2}+8^{2}=x^{2}\)

\(\displaystyle \dpi{100} \small 36+64=x^{2}\) 

\(\displaystyle \dpi{100} \small x=10\) miles

\(\displaystyle The\;hypotenuse\;can\;be\;between\;0\;and\;1.\)

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:

\(\displaystyle a^2 + b^2 = c^2\)

\(\displaystyle 5^2 + 12^2 = c^2\)

\(\displaystyle c^2 = 25 + 144 = 169\)

\(\displaystyle c= \sqrt{169} = 13\)

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