Question

The side lengths of a 30-60-90 triangle are in the ratio 1: 3: 2. What is tan 30°? V3 3 B. 1 2 C. 3 2 O D. 3 SUBMIT

Answer 1

The correct answer is: Option A

Problem Statement:

The question says we have a 30-60-90 triangle and we are to determine the value of tan 30 degrees.

The ratio of the lengths is given as:

`1\colon\sqrt[]{3}\colon2`

Method:

In order to solve this problem, we need to:

1. Visualize the problem using a sketch

2. Apply the SOHCAHTOA theorem to the triangle to find the value of tan 30 degrees.

Implementation:

1. Visualize the problem using a sketch:

Below is a sketch of the triangle given in the question:

The way the lengths of the triangle are arranged is based on proportionality with the angles. Each angle is equal

in proportion relative to the length that is opposite it.

So, because 30 degrees is the smallest angle, the opposite length must be the shortest length i.e. 1. Also, because

90 degrees is the largest angle in the triangle, the opposite length to the angle must be 2 since 2 is the largest length.

2. Apply SOHCAHTOA:

The triangle lengths are related to the angles using SOHCAHTOA. Since we have been told to find tan 30, it means that

we need to make 30 degrees the base angle. Right now, in the orientation above, 60 degrees is the base angle.

We can redraw the triangle and apply SOHCAHTOA as shown below:

I have taken the liberty of labelling the triangle for easier nomenclature.

AB is the Hypotenuse, CB is the Adjacent and AC is the Opposite.

So to find tan 30, we use the "TOA" portion of SOHCAHTOA as shown below:

`\begin{gathered} \text{TOA corresponds to:} \\ \tan \theta=\frac{\text{Opposite}}{\text{Adjacent}} \\ \theta=\text{base angle} \\ \\ \text{Opposite}=AC=1 \\ \text{Adjacent}=CB=\sqrt[]{3} \\ \theta=30^0 \\ \\ \tan 30^0=\frac{1}{\sqrt[]{3}} \\ \text{Rationalizing the answer, we have:} \\ \tan 30^0=\frac{1}{\sqrt[]{3}}*\frac{\sqrt[]{3}}{\sqrt[]{3}} \\ \\ \therefore\tan 30^0=\frac{\sqrt[]{3}}{3} \end{gathered}`

Therefore, the final answer is:

Option A