Lesson Plan - 30 60 90 Triangle
We begin with the equilateral ΔABC, with side lengths of 2.
We need to show that the angle bisector of ∠CBA is the perpendicular bisector of CA.
Note that since ABC is an equilateral triangle, and all three sides are equal, then it is also an isosceles triangle. This is because any two sides of the triangle are equal.
One property of an angle bisector is that the two segments that form the angle are equidistant from the angle bisector. Now since ΔABC is isosceles then CB≡BA, and points C and A are equidistant from point B. By the angle bisector property that we just discussed, C and A are then equidistant from the angle bisector, meaning they are equidistant from point D. Since D is on segment CA, then this means that segment CD is the same length as DA. Therefore, point D cuts segment CA in half.
So now we have the bisector part, all we need to show is that BD is perpendicular to CA, then we will be able to show that ΔBDC is half of ΔABC. Now I would like to bring in the Perpendicular Bisector Theorem. This theorem states that if any point is equidistant from the endpoints of a given segment, then it lies on the perpendicular bisector. Given that ΔABC is isosceles, then point B is equidistant from the endpoints of AC. But since AD is the angle bisector of ∠CBA, then every point on the angle bisector lies equidistant from the endpoints of AC! Thus, the bisector is both an angle bisector, and the perpendicular bisector of CA.
Alright, so what does all this mean?
Well since BD perpendiclarly bisects CA, then ΔCDB is a right triangle and ΔADB is a right triangle! Since CD is congruent to AD, the triangles share a side, and the angles in between are 90 degrees, then the two triangles are congruent by SAS. Anyways, lets turn our attention to triangle ΔADB. I have recopied and relabeled it as ΔA1EB1.
Now since ΔABC had side lengths of 2, then B1A1 would have side length 2. Since D bisected CA, then AD had length half of the full length of that side, meaning the length of AD is 1. Thus, EA1 has length of 1. We are so close!
Since ΔA1EB1 is a right triangle, then we can use the Pythagorean Theorem!
Applying this to find the length of B1E, we have
Now we have it! This triangle is very special. We call this a 30-60-90 Triangle because of its three angles: the right angle, the 60 degree angle left over from the fact that this is half of an equilateral triangle, and the 30 degree angle that was a 60 degree angle split in two by the angle bisector.
To see the special nature of this triangle, we inspect the ratio of its sides:
Notice that if we stretch the triangle to any similar size, say 5 times bigger or so, then the ratio will persist!
The ratio of this triangle would be
And if we divide each side length by the rate of expansion, then we get our original ratio again!
This is spectacular, because for any 30-60-90 Triangle, the ratio of the sides will look like so:
We can use this powerful knowledge to find the missing side length of any triangle given one side and the fact that its a 30-60-90 Triangle.
Amazing!