In the figure to the left, what is the measure of c?
Seeing a right triangle, your first instinct might be to try the Pythagorean theorem. However, with the measure of only one side, we can’t use this method to find c. Our next hope could be to use our trigonometric functions. Using our graphing calculator and SOH-CAH-TOA, we can enter sin(30) (making sure we’re in degree mode!), giving us a value of 0.5. since sin = opposite/hypotenuse, we know that c must be 8, and 4/8 = 0.5. However, there is another way we can approach this problem, one that can save us A LOT of time!
This triangle, with angles of 30, 60, and 90 degrees, is a special kind of right triangle with specific properties you should be familiar with.
In a 30-60-90 triangle, the sides follow the pattern in the figure above. If the shortest side—the side opposite the 30-degree angle—is x, then the measure of the other side is √3 * x, and the hypotenuse measures 2x. So, in our original problem, the shortest side is 4, so we know that the hypotenuse is two times that: 8. How simple was that? Pretty darn, if you ask us! By taking one look at the figure and doing one simple calculation, we solved the problem and shaved off precious time from our total test.
The other special right triangle you should be familiar with before taking the ACT or SAT is the other one in the figure above: the 45-45-90 triangle. If the measure of the non-hypotenuse sides is x, then the measure of the hypotenuse is √2 * x. So, if our original problem asked to find c for a 45-45-90 triangle instead, our answer would be 4√2 (about 5.65).
