The difference of the areas of two squares drawn on two line segments of different lengths is 32 sq. cm. Find the length of the greater line segment if one is longer than the other by 2 cm.

Options

A. 7 cm
B. 9 cm
C. 11 cm
D. 16 cm

Answer

To solve the problem, we will follow these steps:

1. **Define the Variables**:
Let the length of the shorter line segment be \( x \) cm. Since the other line segment is 2 cm longer, the length of the longer line segment will be \( x + 2 \) cm.

2. **Calculate the Areas of the Squares**:
The area of the square drawn on the shorter line segment is:
\[ \text{Area}_{\text{short}} = x^2 \text{ sq. cm} \]
The area of the square drawn on the longer line segment is:
\[ \text{Area}_{\text{long}} = (x + 2)^2 \text{ sq. cm} \]

3. **Set Up the Equation**:
According to the problem, the difference in the areas of the two squares is 32 sq. cm. Therefore, we can write the equation:
\[ (x + 2)^2 - x^2 = 32 \]

4. **Expand the Equation**:
Expanding the left side:
\[ (x^2 + 4x + 4) - x^2 = 32 \]
This simplifies to:
\[ 4x + 4 = 32 \]

5. **Solve for \( x \)**:
Subtract 4 from both sides:
\[ 4x = 28 \]
Now, divide both sides by 4:
\[ x = 7 \text{ cm} \]

6. **Find the Length of the Greater Line Segment**:
The length of the greater line segment is:
\[ x + 2 = 7 + 2 = 9 \text{ cm} \]

Thus, the length of the greater line segment is **9 cm**.