The image shows a right triangle with one leg labeled as $y$, the other as $9$, and the hypotenuse as $17$. The area of the triangle is given as $54 \, \text{units}^2$, and we need to find the length of $y$.

Approach:

1. The formula for the area of a right triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ In this case, the base is $9$ and the height is $y$, so: $\text{Area} = \frac{1}{2} \times 9 \times y$

2. The area is given as $54 \, \text{units}^2$, so: $54 = \frac{1}{2} \times 9 \times y$

3. Solve for $y$: $54 = \frac{9y}{2}$ Multiply both sides by 2: $108 = 9y$ Divide by 9: $y = \frac{108}{9} = 12$

Thus, the length of $y$ is $\boxed{12}$.

Would you like more details or have any questions?

Here are five related questions to explore further:

1. What is the formula to calculate the hypotenuse of a right triangle using the Pythagorean theorem?
2. How would you find the perimeter of this triangle?
3. If the area was given as 81, how would the length of $y$ change?
4. What is the relationship between the area and the sides of a triangle?
5. Can you apply the concept of trigonometry to find angles in this triangle?

Tip: Always double-check the units when solving geometry problems to ensure accuracy!