The problem presents a right triangle with one side measuring 108 units and the diagonal (hypotenuse) measuring 117 units. We are asked to find the area of the right triangle.

Approach:

Let's denote the right triangle's vertices as:

• $A$ and $B$ being the two perpendicular sides.
• $C$ being the point where the hypotenuse $AC$ meets the right angle at $B$.

Given:

• One side (leg) $AB = 108$
• The diagonal (hypotenuse) $AC = 117$

We can use the Pythagorean Theorem to find the missing side $BC$. The Pythagorean theorem states:

$AC^2 = AB^2 + BC^2$

Substitute the known values:

$117^2 = 108^2 + BC^2$

Calculation:

$117^2 = 13689, \quad 108^2 = 11664$

Now, solve for $BC$:

$13689 = 11664 + BC^2$ $BC^2 = 13689 - 11664 = 2037$ $BC = \sqrt{2037} \approx 45.16$

So, the length of side $BC$ is approximately 45.16.

Area of the right triangle:

The area $A$ of a right triangle is given by:

$A = \frac{1}{2} \times \text{base} \times \text{height}$

Here, base = $AB = 108$ and height = $BC \approx 45.16$.

$A = \frac{1}{2} \times 108 \times 45.16 = 2434.32$

Thus, the area of the triangle is approximately 2434.32 square units.

Let me know if you need further details or have any questions!

Here are 5 related questions to expand your understanding:

1. How does the Pythagorean theorem help in calculating the unknown side of a right triangle?
2. What would happen if the given triangle was not a right triangle? How would the area be calculated?
3. How do you calculate the perimeter of a right triangle given all sides?
4. What is the significance of the hypotenuse in a right triangle, and how does it relate to the other sides?
5. How would you solve for the area of a rectangle if one side is given along with the diagonal?

Tip: When applying the Pythagorean theorem, always check if the given triangle is a right triangle first!