Find the length of diagonals of a rectangle AOBC whose three vertices are A(0, 3),O(0, 0) and B(5, O).

Options

A. `sqrt23` units
B. 5 units
C. `sqrt21` units
D. `sqrt(34)units

Answer

To find the length of the diagonals of rectangle AOBC with vertices A(0, 3), O(0, 0), and B(5, 0), we first need to determine the coordinates of the fourth vertex, C.

### Step-by-Step Solution:

1. **Identify the coordinates of the vertices:**
- A(0, 3)
- O(0, 0)
- B(5, 0)

2. **Determine the coordinates of vertex C:**
Since AOBC is a rectangle, the coordinates of C can be found using the fact that opposite sides are equal and parallel. The coordinates of C will be:
- C(5, 3) (since it shares the x-coordinate with B and the y-coordinate with A).

3. **Use the distance formula to find the length of the diagonals:**
The length of the diagonal can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We can calculate the diagonal AO or OC (both will be the same length).

4. **Calculate the length of diagonal AO:**
- For diagonal AO, we have:
- A(0, 3) and O(0, 0)
- Using the distance formula:
\[ AO = \sqrt{(0 - 0)^2 + (3 - 0)^2} = \sqrt{0 + 9} = \sqrt{9} = 3 \]

5. **Calculate the length of diagonal OB:**
- For diagonal OB, we have:
- O(0, 0) and B(5, 0)
- Using the distance formula:
\[ OB = \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{25 + 0} = \sqrt{25} = 5 \]

6. **Calculate the length of diagonal AC:**
- For diagonal AC, we have:
- A(0, 3) and C(5, 3)
- Using the distance formula:
\[ AC = \sqrt{(5 - 0)^2 + (3 - 3)^2} = \sqrt{25 + 0} = \sqrt{25} = 5 \]

7. **Calculate the length of diagonal BC:**
- For diagonal BC, we have:
- B(5, 0) and C(5, 3)
- Using the distance formula:
\[ BC = \sqrt{(5 - 5)^2 + (3 - 0)^2} = \sqrt{0 + 9} = \sqrt{9} = 3 \]

8. **Conclusion:**
The lengths of the diagonals AO and OC are equal, and the lengths of diagonals AC and BC are also equal. Therefore, the length of the diagonals of rectangle AOBC is:
\[ \sqrt{34} \text{ units} \]