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

Options

A. `sqrt23` units
B. 5 units
C. `sqrt21` units
D. `sqrt34` 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 can follow these steps:

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

### Step 2: Determine the fourth vertex C
Since AOBC is a rectangle, we can find the coordinates of the fourth vertex C. The coordinates of C can be found by using the fact that opposite sides of a rectangle are equal and parallel. The coordinates of C can be determined as follows:
- C will have the same x-coordinate as B and the same y-coordinate as A.
Thus, C = (5, 3).

### Step 3: Use the distance formula to find the length of diagonal AB
The length of diagonal AB can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, A(0, 3) and B(5, 0) are the two points.

Substituting the coordinates into the formula:
\[ d_{AB} = \sqrt{(5 - 0)^2 + (0 - 3)^2} \]
\[ = \sqrt{5^2 + (-3)^2} \]
\[ = \sqrt{25 + 9} \]
\[ = \sqrt{34} \]

### Step 4: Find the length of diagonal OC
Now, we can find the length of the diagonal OC using the same distance formula:
\[ d_{OC} = \sqrt{(x_C - x_O)^2 + (y_C - y_O)^2} \]
Here, O(0, 0) and C(5, 3) are the two points.

Substituting the coordinates into the formula:
\[ d_{OC} = \sqrt{(5 - 0)^2 + (3 - 0)^2} \]
\[ = \sqrt{5^2 + 3^2} \]
\[ = \sqrt{25 + 9} \]
\[ = \sqrt{34} \]

### Final Answer
The lengths of both diagonals AB and OC are \(\sqrt{34}\) units.

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