Area and Side Relationship in a Right Triangle

In triangle ABC, angle A measures $90^{ ext{o}}$ and angle B measures $30^{ ext{o}}$. The length of side AB is $x$ units, and side AC is $y$ units. If the area of triangle ABC is 60 square units, what is the value of $x + y$?

Recall that in a right triangle, the area can be calculated as:

Area = $\f\frac{1}{2} \times \text{base} \times \text{height}$.

In this case, if we take AB as the base and AC as the height, the area can also be expressed as $\f\frac{1}{2} \times x \times y = 60$. Hence, we have:

$$ x \cdot y = 120. $$

Using the properties of the 30-60-90 triangle, we know:

1. $BC = 2 \cdot AC$ (side opposite 30 degrees)
2. $AB = \sqrt{3} \cdot AC$ (side opposite 60 degrees)