1. In a 30-60-90 triangle, the length of the hypotenuse is 6. What is the length of the shortest side? 2. What is the area of a circle with a diameter of 16?

Steps to Solve

1. Identify sides of a 30-60-90 triangle

In a 30-60-90 triangle, the relationships between the lengths of the sides are given by integers: the shortest side (opposite the 30° angle) is $x$, the longer leg (opposite the 60° angle) is $x\sqrt{3}$, and the hypotenuse is $2x$.

2. Using the hypotenuse to find the shortest side

Given that the hypotenuse is 6, we can set up the equation: $$ 2x = 6 $$ To find $x$, divide both sides by 2: $$ x = \frac{6}{2} = 3 $$

3. Conclusion about the shortest side

The shortest side of the triangle, which corresponds to the value of $x$, is thus 3.

4. Calculate the area of the circle

For the second question, use the formula for the area of a circle: $$ \text{Area} = \pi r^2 $$ First, find the radius. The diameter is given as 16, so the radius $r$ is: $$ r = \frac{16}{2} = 8 $$

5. Substitute the radius into the area formula

Now substitute the radius into the area formula: $$ \text{Area} = \pi (8)^2 = \pi \cdot 64 = 64\pi $$