Are the points (1, 7), (4, 2), (-1, -1) and (-4, 4) vertices of a square ? If yes, what is the length of the side of square ?

Options

A. `34`
B. `sqrt34`
C. `-34`
D. `68`

Answer

To determine if the points (1, 7), (4, 2), (-1, -1), and (-4, 4) are the vertices of a square, we will follow these steps:

### Step 1: Identify the Points
Let:
- A = (1, 7)
- B = (4, 2)
- C = (-1, -1)
- D = (-4, 4)

### Step 2: Calculate the Distances Between Each Pair of Points
We will calculate the distances between each pair of adjacent points (AB, BC, CD, DA) and check if they are equal.

**Distance Formula:**
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Calculate AB
Using points A and B:
\[ AB = \sqrt{(4 - 1)^2 + (2 - 7)^2} = \sqrt{(3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

### Step 4: Calculate BC
Using points B and C:
\[ BC = \sqrt{(-1 - 4)^2 + (-1 - 2)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

### Step 5: Calculate CD
Using points C and D:
\[ CD = \sqrt{(-4 - (-1))^2 + (4 - (-1))^2} = \sqrt{(-3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

### Step 6: Calculate DA
Using points D and A:
\[ DA = \sqrt{(1 - (-4))^2 + (7 - 4)^2} = \sqrt{(5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

### Step 7: Compare the Distances
We have:
- \(AB = \sqrt{34}\)
- \(BC = \sqrt{34}\)
- \(CD = \sqrt{34}\)
- \(DA = \sqrt{34}\)

Since all four sides are equal, we can conclude that the points form a square.

### Step 8: Determine the Length of the Side of the Square
The length of the side of the square is:
\[ \text{Length of side} = \sqrt{34} \]

### Conclusion
Yes, the points (1, 7), (4, 2), (-1, -1), and (-4, 4) are the vertices of a square, and the length of the side of the square is \(\sqrt{34}\).

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