A 30-60-90 triangle is a right triangle. Triangles with a right angle are called right triangles. A right triangle can only have a right angle. The representation of this problem is shown below. So let's demonstrate why the mentioned options are correct:
The hypotenuse of a right triangle is always opposite to the right angle. If we name [tex]a[/tex] as the shorter leg, for the sine of law is true that the hypotenuse is:
[tex]\frac{H}{sin90}=\frac{a}{sin30} \\ \\ \frac{H}{1}=\frac{a}{0.5} \\ \\ \boxed{H=2a}[/tex]
So this fact tells us that the hypotenuse is twice as long as the shorter leg
The longer leg, let's name it [tex]x[/tex] can be calculated using Pythagorean Theorem:
[tex]H^2=x^2+a^2 \\ \\ x=\sqrt{H^2-a^2} \\ \\ x=\sqrt{(2a)^2-a^2} \\ \\ x=\sqrt{4a^2-a^2} \\ \\ x=\sqrt{3a^2}=\sqrt{3}a[/tex]
So it is true that the longer leg is √3 times as long as the shorter leg.
Answer:
The longer leg is /3 times as long as the shorter leg.
The hypotenuse is twice as long as the shorter leg.
Step-by-step explanation:
Given triangle is 30- 60-90 degree triangle
WE use the common ratio of 30- 60-90 degree triangle
Common ratio is 1:1[tex]\sqrt{3}[/tex]:2
30 : 60 : 90
Short leg : long leg : hypotenuse
x : [tex]x\sqrt{3}[/tex] : 2x
The hypotenuse is 2 times of the shorter leg
longer leg is square root (3) times the shorter leg
The longer leg is /3 times as long as the shorter leg.
The hypotenuse is twice as long as the shorter leg.
