Module 9: Geometry

As we’ve seen in the previous section, there is an inherent directionality involved with right triangle trigonometry. The same concept applies to cardinal directions, otherwise known as north, south, east, west.

For example, imagine you are on a sailboat traveling north across the width of a river to get to a small dock on the other side. However, the current of the river (moving east) is hindering the boat from traveling strictly north to the dock. When you finally reach the other side of the lake, you notice you are 28 m out from the docks. If you travelled 56 m of distance during your trip, how wide is the lake?

First, it’s important to keep track of all the necessary information and use a diagram to provide visual clarity. Since it is known that the dock is directly north of the starting point, the stopping point is 28 m out from the dock, and that the boat travelled 56 m given a current moving east, we can assume the boat stopped east of the dock and travelled diagonally given the current. With that, we can model the scenario below:

To determine the width of the river, we can employ the Pythagorean theorem. Since the hypotenuse is known, we’ll need to rearrange and solve for the leg instead:

$${"version":"1.1","math":"a^2+b^2=c^2"}$$

$${"version":"1.1","math":"a^2=c^2-b^2"}$$

$${"version":"1.1","math":"a^2=\sqrt{c^2-b^2}"}$$

$${"version":"1.1","math":"a^2=\sqrt{56^2-28^2}"}$$

$${"version":"1.1","math":"a\cong48.5"}$$

Alternatively, notice the lengths of the known sides when comparing them to any of the special cases. Recall that the special case triangles hold true for any triangle that share proportional side lengths. Here, the hypotenuse and shorter arm, being 56 m and 28 m, are proportional to the respective sides from the 30-60-90 special case, being 2 m and 1 m. The scaling factor here is 28. Thus, we can determine the third side length by multiplying 28 to ${"version":"1.1","math":"\sqrt3"}$

$${"version":"1.1","math":"28\times \sqrt3=28\sqrt3\cong48.5"}$$

Using both methods provide the same answer. The special cases are a shortcut.

Based off the special case, we can also determine that the angle of trajectory of the boat given he eastward current is 30°. To note this, it can be written as N 30° E. This is a way of signifying direction relative to the cardinal directions. The direction notation of N 30° E translates to north but 30° east of it. This notation may differ, but ultimately means the same thing.