m12, #6

[mehrasa]

If in a right triangle, the ratio of the shortest side to the longest is [m]\frac{1}{2}[/m] , what is the smallest angle in this triangle?
Ans:30

I can not understand why the answer is.
As I know in the triangle, the shortest side is opposite smallest angle and vice verse.. when it is said it is right triangle so the largest angle is 90 and the largest side is diagonal.. here the ratio of smallest to largest is 1/2 so the '2' refer to diagonal which is 90 degree therefore, the smallest is 45 degree.

any further explanation?

[Anurag@Gurome]

This can be easily proved by trigonometry. The information given is equivalent to saying that sine of the smallest angle is equal to 1/2. Which is only possible if the measure of smallest angle is 30 degrees.


We can also conclude this using the famous property of '30-60-90' triangles. Which can be shown as follows.



Refer to the figure above.

Largest side = c and smallest side = a
Thus, (a/c) = 1/2 => c = 2a

Again, (a² + b²) = c² = (2a)² = 4a²
=> b² = 3a²
=> b = (√3)a

Therefore ratio of the three sides of the triangle is a : b : c = a : (√3)a : 2a = 1:√3:2
The ratio of the lengths of three sides of a right-angled triangle is 1:√3:2, only when the triangle is 30-60-90 triangle. Hence the measure of the smallest angle is 30 degrees.

[Night reader]

the longest side is hypotenuse, as such two sides would be non-hypotenuse. The shortest side is always opposite the smallest interior angle. Of the two non-hypotenuse sides the shortest one would be opposite to hypotenuse. The ratio is 1/2 OR non-hypotenuse/hypotenuse. This is relationship 30`-60`-90` or 30` opposite to the shortest side.