Find the distance between the following paris of points :
`(sqrt(3) + 1,1) and (0,sqrt(3))`

Answer

To find the distance between the points \((\sqrt{3} + 1, 1)\) and \((0, \sqrt{3})\), we will use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]

### Step 1: Identify the coordinates
Let:
- Point 1: \((x_1, y_1) = (\sqrt{3} + 1, 1)\)
- Point 2: \((x_2, y_2) = (0, \sqrt{3})\)

### Step 2: Substitute the coordinates into the distance formula
Now, substituting the coordinates into the distance formula:

\[ d = \sqrt{((\sqrt{3} + 1) - 0)^2 + (1 - \sqrt{3})^2} \]

### Step 3: Simplify the expression
This simplifies to:

\[ d = \sqrt{(\sqrt{3} + 1)^2 + (1 - \sqrt{3})^2} \]

### Step 4: Expand the squares
Now we will expand both squares:

1. \((\sqrt{3} + 1)^2 = (\sqrt{3})^2 + 2(\sqrt{3})(1) + (1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}\)
2. \((1 - \sqrt{3})^2 = (1)^2 - 2(1)(\sqrt{3}) + (\sqrt{3})^2 = 1 - 2\sqrt{3} + 3 = 4 - 2\sqrt{3}\)

### Step 5: Combine the results
Now, combine the results from the expansions:

\[ d = \sqrt{(4 + 2\sqrt{3}) + (4 - 2\sqrt{3})} \]

This simplifies to:

\[ d = \sqrt{4 + 4} = \sqrt{8} \]

### Step 6: Simplify \(\sqrt{8}\)
We can simplify \(\sqrt{8}\):

\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \]

### Final Answer
Thus, the distance between the points \((\sqrt{3} + 1, 1)\) and \((0, \sqrt{3})\) is:

\[ d = 2\sqrt{2} \]