A rhombus and a square have the same base. If the diagonals of the rhombus measure 30 cm and 16 cm respectively, find the area of the square.

Options

A. 225 `cm^(2)`
B. 200 `cm^(2)`
C. 240 `cm^(2)`
D. 289 `cm^(2)`

Answer

To find the area of the square given that a rhombus and a square have the same base, and the diagonals of the rhombus measure 30 cm and 16 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths of the diagonals of the rhombus**:
- The lengths of the diagonals are given as 30 cm and 16 cm.

2. **Calculate half of each diagonal**:
- Half of the first diagonal (d1) = 30 cm / 2 = 15 cm
- Half of the second diagonal (d2) = 16 cm / 2 = 8 cm

3. **Use the properties of the rhombus**:
- The diagonals of a rhombus bisect each other at right angles. Therefore, we can form a right triangle with half of each diagonal as the legs.
- The lengths of the legs of the triangle are 15 cm and 8 cm.

4. **Apply the Pythagorean theorem to find the side of the rhombus**:
- Let the side of the rhombus be \( y \).
- According to the Pythagorean theorem:
\[ y^2 = (15)^2 + (8)^2 \]
- Calculate \( (15)^2 \) and \( (8)^2 \):
\[ y^2 = 225 + 64 \]
- Add the two results:
\[ y^2 = 289 \]

5. **Find the value of \( y \)**:
- Taking the square root of both sides:
\[ y = \sqrt{289} = 17 \text{ cm} \]

6. **Calculate the area of the square**:
- Since the side of the square is equal to the side of the rhombus, the area of the square (A) is given by:
\[ A = y^2 = (17)^2 = 289 \text{ cm}^2 \]

### Final Answer:
The area of the square is **289 cm²**.