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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

AThe second-longest side is twice as long as the shotest side.
BThe longest side is twice as long as the shortest side.
CTwo sides of the triangle have the same length
DThe longest side is square root 3 times as long as the shorted side

Verified Answer :

calculista

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

The correct statement is :

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

Step-by-step explanation:

Given information:

A triangle has a \(30^o\) and a \(60^o\) angle

Now , the third angle will be \(90^o\) angle.

So, it is a right angle triangle:

\(sin (30^o)=cos(60^o)=1/2\\cos (306o)=sin (60^o)=\sqrt{3}/2\)

Now, check the 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 longest side is \(\sqrt{3} /1\)

Hence, the statement is false.

(B)The longest side is twice as long as the shortest side

As we know

The ratio between the longest side and shortest side is 2/1

Hence, the statement is true.

(C)Two sides of triangles have same length

The triangle give is not an isosceles triangle

So, the statement is false.

(D)The longest side is \(\sqrt{3}\) times the shortest side:

The ratio between the longest side and shortest side is 2.

Hence , the statement is false.

Hence the correct statement is :

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

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