The points (2, 3), (5, 8), (0,5) and (-3, 0) are the vertices of a

Options

A. square
B. rectangle
C. parallelogram
D. rhombus

Answer

To determine the type of quadrilateral formed by the points (2, 3), (5, 8), (0, 5), and (-3, 0), we will calculate the lengths of the sides and the diagonals using the distance formula.

### Step-by-Step Solution:

1. **Label the Points:**
Let:
- A = (2, 3)
- B = (5, 8)
- C = (0, 5)
- D = (-3, 0)

2. **Calculate the Length of Side AB:**
\[ AB = \sqrt{(5 - 2)^2 + (8 - 3)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \]

3. **Calculate the Length of Side BC:**
\[ BC = \sqrt{(0 - 5)^2 + (5 - 8)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

4. **Calculate the Length of Side CD:**
\[ CD = \sqrt{(0 + 3)^2 + (5 - 0)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \]

5. **Calculate the Length of Side AD:**
\[ AD = \sqrt{(2 + 3)^2 + (3 - 0)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \]

6. **Calculate the Length of Diagonal AC:**
\[ AC = \sqrt{(2 - 0)^2 + (3 - 5)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \]

7. **Calculate the Length of Diagonal BD:**
\[ BD = \sqrt{(5 + 3)^2 + (8 - 0)^2} = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \]

8. **Compare the Lengths:**
- All sides (AB, BC, CD, AD) are equal: \( AB = BC = CD = AD = \sqrt{34} \)
- The diagonals are not equal: \( AC = \sqrt{8} \) and \( BD = 8\sqrt{2} \)

Since all sides are equal and the diagonals are not equal, the quadrilateral formed by the points is a **Rhombus**.