The lengths of the sides and segments of the right triangle representing the distances in the race path are;
2. 15 miles
3. a. 5.4 miles
b. 9.6 miles
4. 7.2 miles
The steps that can be used to obtain the distances between the specified locations on on the race paths are;
2. The distance from the start to finish is the hypotenuse of the large right triangle
According to the Pythagorean theorem, we get;
Direct distance from the start to finish is;√(12² + 9²) = 15 miles
3. a. Let d represent the distance from the Start to the 2nd Buoy
The geometric mean theorem indicates that we get;
9² = d × The direct distance from the start to finish
Therefore;
9² = d × 15
d = 9²/15
9²/15 = 5.4
d = 5.4 miles
b. The sum of the distance from the start to the second Buoy, d, and the distance from the finish to the second Buoy is equivalent to the direct distance from the Start to finish, therefore;
Let l represent the distance from 2nd buoy to the finish line
d + l = 15
5.4 + l = 15
Distance from 2nd buoy to the finish line, l is; 15 - 5.4 = 9.6 miles
4. The distance between the two judges stationed at the two buoys can be obtained from the following geometric mean formula;
Let h represent the distance between the two judges stationed on the two buoys, we get;
h² = d × l
h² = 5.4 × 9.6
5.4 × 9.6 = 51.84
h = √(51.84)
h = 7.2 miles
The distance between the two judges is 7.2 miles