https://file.aiquickdraw.com/mathbot/file/1725713246072bj7zolx4.jpg

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!

The area of the triangle below is 54 units^2. What is y?

Discover how to calculate the length of a leg in a right triangle when the area and one leg are known. This problem uses the area formula for right triangles and provides a clear example of applying basic geometric principles. Suitable for students in Grades 6-8.

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