Question

Which of the following are true statements about a 30-60-90 triangle? Check all that apply.

A. The hypotenuse is twice as long as the longer leg.
B. The longer leg is /3 times as long as the shorter leg.
C. The hypotenuse is twice as long as the shorter leg.
D. The longer leg is twice as long as the shorter leg.
E. The hypotenuse is /3 times as long as the longer leg.
F. The hypotenuse is /3 times as long as the shorter leg.

Answer 1

The longer leg is /3 times as long as the shorter leg.

The hypotenuse is twice as long as the shorter leg.

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
`√(3)`:2

30 : 60 : 90

Short leg : long leg : hypotenuse

x :
`x√(3)` : 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.

Answer 2

C. The hypotenuse is twice as long as the shorter leg.

B. The longer leg is √3 times as long as the shorter leg.

Explanation:

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
`a` as the shorter leg, for the sine of law is true that the hypotenuse is:

`(H)/(sin90)=(a)/(sin30) \\ \\ (H)/(1)=(a)/(0.5) \\ \\ \boxed{H=2a}`

So this fact tells us that the hypotenuse is twice as long as the shorter leg

The longer leg, let's name it
`x` can be calculated using Pythagorean Theorem:

`H^2=x^2+a^2 \\ \\ x=√(H^2-a^2) \\ \\ x=√((2a)^2-a^2) \\ \\ x=√(4a^2-a^2) \\ \\ x=√(3a^2)=√(3)a`

So it is true that the longer leg is √3 times as long as the shorter leg.