Question

A certain triangle has a 30 degree angle and a 60 degree angle. Which of the following must be a true statement about the triangle
A. The second-longest side is twice as long as the shotest side.
B. The longest side is twice as long as the shortest side.
C. Two sides of the triangle have the same length
D. The longest side is square root 3 times as long as the shorted side

Answer 1

we know that

If the triangle has a $30$ degree angle and a $60$ degree angle, then the third angle is a $90$ degree angle

so

is a right triangle

$Sin(30\°)=Cos(60\°)=\frac{1}{2}$

$Cos(30\°)=Sin(60\°)=\frac{\sqrt{3}}{2}$

Statements

A. The second-longest side is twice as long as the shortest side.

we know that

the ratio between the second-longest side and the shortest side is

$\frac{\sqrt{3}}{1}$

therefore

the statement A is False

B. The longest side is twice as long as the shortest side

we know that

the ratio between the longest side and the shortest side is

$\frac{2}{1}$

therefore

the statement B is True

C. Two sides of the triangle have the same length

The statement C is False, because is not an isosceles triangle (45-90-45)

D. The longest side is square root 3 times as long as the shorted side

  we know that

the ratio between the longest side and the shortest side is

$\frac{2}{1}$

The longest side is twice as long as the shortest side

therefore

The statement case D is False

the answer is the option

B. The longest side is twice as long as the shortest side

Answer 2

Answer:

Option B.

Step-by-step explanation:

In the figure attached,

We apply the sine rule in the triangle ABC.

sin30 = $\frac{BC}{AC}$

$\frac{1}{2}=\frac{y}{AC}$

AC = 2y

AC = Hypotenuse or longest side

y = shortest side of the triangle [Since opposite side of the shortest angle]

Therefore, longest side is twice as long as the shortest side.

Option B is the answer.