https://file.aiquickdraw.com/mathbot/file/1727433405058al2kqnif.jpg

To find the length \( y \) in the figure, let's analyze the shape. The diagram shows a trapezoid divided into two parts: a triangle on the left and a square on the right.

### Step 1: Breakdown of the areas
- The **square** on the right has a side length of \( y \). Thus, the area of the square is:
  \[
  \text{Area of the square} = y^2
  \]
  
- The **triangle** on the left is a right triangle with a base and height both equal to \( y \). Thus, the area of the triangle is:
  \[
  \text{Area of the triangle} = \frac{1}{2} \times y \times y = \frac{y^2}{2}
  \]
  
### Step 2: Total area of the figure
The total shaded area is the sum of the area of the square and the triangle, which is given as 120 square inches:
\[
\text{Total shaded area} = y^2 + \frac{y^2}{2} = 120
\]

### Step 3: Solve the equation
We can combine the terms on the left side:
\[
y^2 + \frac{y^2}{2} = \frac{2y^2}{2} + \frac{y^2}{2} = \frac{3y^2}{2}
\]
Now, set this equal to 120 and solve for \( y \):
\[
\frac{3y^2}{2} = 120
\]
Multiply both sides by 2 to eliminate the fraction:
\[
3y^2 = 240
\]
Now, divide by 3:
\[
y^2 = 80
\]
Finally, take the square root of both sides:
\[
y = \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \approx 8.94
\]

Thus, the length \( y \) is approximately **8.94 inches**.

Would you like a detailed explanation of any step?

### Related Questions:
1. How do you calculate the area of a trapezoid when it isn’t broken into simpler shapes?
2. If the total area was given as 150 in², how would the length \( y \) change?
3. What is the formula for the area of a right triangle, and how can it be applied to different problems?
4. How does the Pythagorean theorem relate to the area of squares and triangles?
5. Can we calculate \( y \) if the shaded area includes additional non-geometrical shapes?

**Tip:** When solving geometry problems, always break the complex shapes into simple ones (like triangles and squares) to make calculations easier.

Find the length y in the figure, if the shaded area is 120 in².

This geometry problem explores how to find the length y in a figure consisting of a square and a triangle, with a total shaded area of 120 in². Learn step-by-step how to use area formulas for squares and triangles to solve for y. Suitable for Grades 7-10.

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