If AOBC is a rectangle whose three vertices are A(0,3),O(0,0) and B(5,0), then the length of its diagonal is

Options

A. 5
B. 3
C. `sqrt(34)`
D. 4

Explanation

To find the length of the diagonal of rectangle AOBC given the vertices A(0,3), O(0,0), and B(5,0), we can follow these steps:

### Step 1: Identify the coordinates of the points
We have the coordinates of three vertices:
- A(0, 3)
- O(0, 0)
- B(5, 0)

### Step 2: Determine the coordinates of point C
Since AOBC is a rectangle, point C can be determined using the coordinates of points A and B. The coordinates of point C will be (5, 3) because it shares the x-coordinate with B and the y-coordinate with A.

### Step 3: Use the distance formula to find the length of diagonal AB
The distance formula to find the distance between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

Here, we will calculate the distance between points A(0, 3) and B(5, 0).

### Step 4: Substitute the coordinates into the distance formula
Let:
- \( (x1, y1) = (0, 3) \) (coordinates of A)
- \( (x2, y2) = (5, 0) \) (coordinates of B)

Substituting these values into the distance formula:
\[ d = \sqrt{(5 - 0)^2 + (0 - 3)^2} \]

### Step 5: Calculate the squares
Calculating the squares:
\[ d = \sqrt{(5)^2 + (-3)^2} = \sqrt{25 + 9} \]

### Step 6: Add the results
Adding the results:
\[ d = \sqrt{34} \]

### Step 7: Conclusion
The length of the diagonal AB is \(\sqrt{34}\).