Question

Find a. Write your answer in simplest radical form.

Answer 1

Answer:

$a=6mm$

Skills needed: Special Right Triangles

Step-by-step explanation:

1) We are given what is known as a right triangle. More specifically, this is a special right triangle. It is a:

30-60-90 triangle, which means it has a:

---> 30-degree angle

---> 60-degree angle

---> 90-degree (right) angle

There is an important property with the side lengths of this special right triangle.

2) The property is the following:

- Let's say the side opposite of the 30-degree angle is $b$.

- The side opposite the 90-degree angle would be $2b$.

- The side opposite the 60-degree angle would be $b * \sqrt3$

This property is useful for finding side lengths of the triangle.  

3) In this problem, we are given the side opposite the 90-degree angle, which means we can divide by $2$ to solve for $b$ (side opposite 30° angle).

$4\sqrt{3} = 2b$ ---> This is the equation

$4\sqrt3 \div 2 = b$ --> We divide by 2 on both sides to isolate the variable.

$b = 2\sqrt3$ --> We evaluate it out, and get this to be our answer for $a$

Then, we can solve for the side opposite the 60-degree angle by multiplying $b$ by $\sqrt3$

---> $b*\sqrt3=2\sqrt3*\sqrt3 = 2*\sqrt3*\sqrt3=2*3=6$

$\sqrt3$ times itself would be 3, as multiplying by itself is squaring the square root, and removing it.

---> $a=6mm$ in the diagram.