In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. Why?

Question

> In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. A statement from the trigonometry section of Simmons' *Precalculus in a nutshell*. Please explain.

Answer 1

Draw an equilateral triangle ABC. Angle A,B and C would all be 60 degrees. From vertex B draw a perpendicular bisector to AC and label the point where it intersects AC as point D. This will give you 2 congruent triangles ABD and CBD wnich are both 30-60-90 triangles. Since this line bisects AC, AD is 1/2 of AC. But AC is the same length as AB (remember triangle ABC is equilateral) AB is the hypotenuse of triangle ABD and AD is the side opposite the 30 degree angle. Therefore the side opposite the 30 angle is 1/2 the hypotenuse

Answer 2

Try sketching an equilateral triangle (each angle 60^0, sides equal). Drop an altitude from one vertex to the opposite side. Can you prove that the altitude bisects the base? Using congruent triangles? If so then the altitude also bisects the vertex angle (into two 30 degree angles). The sketch should convince you that you can prove that for one of the two congruent triangles created, the side opposite the 30 is 1/2 the side opposite the right angle (the hypotenuse).

Answer 3

Let x be the side opposite the 30 degree angle, and I will use hyp to represent the hypotenuse.

Then sin 30 = x / hyp (definition of sin function)

hyp (sin 30) = x (mult. both sides by hyp), but, sin 30 = 1/2 so, this gives us

hyp (1/2) = x, so x is 1/2 times the hyp.

Patti D.