Show that the points A (5,6) B (1,5) and C (2,1) and D (6,2) are the vertices of a square ABCD.

Answer

To show that the points A(5, 6), B(1, 5), C(2, 1), and D(6, 2) are the vertices of a square ABCD, we will follow these steps:

### Step 1: Calculate the lengths of the sides of the quadrilateral ABCD using the distance formula.

The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

#### 1.1 Length of AB:
Let’s calculate the length of AB:
\[ AB = \sqrt{(1 - 5)^2 + (5 - 6)^2} = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

#### 1.2 Length of BC:
Now, calculate the length of BC:
\[ BC = \sqrt{(2 - 1)^2 + (1 - 5)^2} = \sqrt{(1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \]

#### 1.3 Length of CD:
Next, calculate the length of CD:
\[ CD = \sqrt{(6 - 2)^2 + (2 - 1)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

#### 1.4 Length of AD:
Finally, calculate the length of AD:
\[ AD = \sqrt{(6 - 5)^2 + (2 - 6)^2} = \sqrt{(1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \]

### Step 2: Check if all sides are equal.
From the calculations above, we have:
- AB = BC = CD = AD = \(\sqrt{17}\)

Since all sides are equal, ABCD is a rhombus.

### Step 3: Calculate the lengths of the diagonals AC and BD.

#### 3.1 Length of AC:
Now, let’s calculate the length of diagonal AC:
\[ AC = \sqrt{(2 - 5)^2 + (1 - 6)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

#### 3.2 Length of BD:
Now, calculate the length of diagonal BD:
\[ BD = \sqrt{(6 - 1)^2 + (2 - 5)^2} = \sqrt{(5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

### Step 4: Check if the diagonals are equal.
From the calculations above, we have:
- AC = BD = \(\sqrt{34}\)

### Conclusion:
Since all sides are equal and both diagonals are equal, quadrilateral ABCD is a square.

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