What is the length of a side of an equilateral triangle whose altitude has a length of 18?

Answer 1

This is going to take some drawing, and a calculator, but first let's get the facts. An equilateral triangle is also equiangular, meaning all angles are 60 degrees. The altitude, meaning starting from a vertex it to a direction so that it is perpendicular to the opposite side, is 18 in length.

OK, now for the drawing. An equiangular triangle with a line through the middle. The line in the middle is the altitude at 18. Now, ignore the entire left side of the triangle (or the right, but for this we're ignoring the left). We know that the angle to the far right is 60 degrees because of the whole triangle being equiangular. The bottom angle at the altitude is 90 degrees because the altitude forms a right angle. Because of this, we know the last angle is 30 degrees, because all angles must add up to be 180 degrees. This is a special type of triangle, 30-60-90. They have two formulas. For the formulas, L is the long side which is the side opposite the 60 degree angle. S is the short side, which is the angle opposite the 30 degree angle. H is the hypotenuse, which is the side opposite the 90 degree angle. The formulas are H = S√3 and L = 2S. The 18 altitude is the long, opposite the 60 degree angle. One of the sides of the whole triangle would be the hypotenuse, opposite the right angle. To find the hypotenuse, we must first find the short, and for that we need the long. We know the long is 18, so the whole formula is H = ((1/2)L)√3 or H = (L/2)√3. The longer way is one formula at a time. Plugged in, the formula is 18 = 2S, S = 9. H = 9√3, H = 15.58845727... With the whole formula, H = ((1/2)18)√3 or H = (L/2)√3, your answer is still 15.58845727... Most teachers will just let you leave it as 9√3.

There's a second way to solve this using sine, cosine, or tangent. (You're calculator must be in degree mode). Back to the drawing of half the triangle, you must use one of the three modes depending on the angle you start at. You cannot start at the right angle, you must start at the 30 or 60. O = opposite, H = hypotenuse, A = adjacent. The three formulas are sin = O/H, cos = A/H, and tan = O/A. In relation to the 30 angle, 18 is the adjacent and the side we need is the hypotenuse, so we use cosine. cos30 = 18/x. when the x is on the bottom, u switch x with the other side of the equal sign. x = 18/cos30. That will give you the answer of 20.78460969... PLEASE KEEP IN MIND THAT SINE, COSINE, AND TANGENT ARE NOT ALWAYS EXACT. THE ANSWER IS CLOSE TO THE OTHER ANSWER WE GOT, SO IT'S GOOD. In relation to the 60 angle, 18 is the opposite and the side we need is hypotenuse, so we use sine. sin60 = 18/x. Again, just switch. x = 18/sin60. This will give you the answer of 20.78460969...