Find the distance between the following paris of points :
`((3)/(5),2) and (-(1)/(5),1(2)/(5))`

Answer

To find the distance between the points \((\frac{3}{5}, 2)\) and \((- \frac{1}{5}, \frac{2}{5})\), we can use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

### Step 1: Identify the points
Let:
- Point A: \((x_1, y_1) = \left(\frac{3}{5}, 2\right)\)
- Point B: \((x_2, y_2) = \left(-\frac{1}{5}, \frac{2}{5}\right)\)

### Step 2: Substitute the coordinates into the distance formula
Substituting the coordinates into the formula, we have:

\[ d = \sqrt{\left(-\frac{1}{5} - \frac{3}{5}\right)^2 + \left(\frac{2}{5} - 2\right)^2} \]

### Step 3: Simplify the x-coordinates
Calculate \(x_2 - x_1\):

\[ -\frac{1}{5} - \frac{3}{5} = -\frac{4}{5} \]

Now square this result:

\[ \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \]

### Step 4: Simplify the y-coordinates
Calculate \(y_2 - y_1\):

\[ \frac{2}{5} - 2 = \frac{2}{5} - \frac{10}{5} = -\frac{8}{5} \]

Now square this result:

\[ \left(-\frac{8}{5}\right)^2 = \frac{64}{25} \]

### Step 5: Combine the results
Now substitute back into the distance formula:

\[ d = \sqrt{\frac{16}{25} + \frac{64}{25}} = \sqrt{\frac{80}{25}} = \sqrt{3.2} \]

### Step 6: Simplify the square root
We can simplify \(\sqrt{3.2}\):

\[ \sqrt{3.2} = \sqrt{\frac{32}{10}} = \frac{4\sqrt{2}}{5} \]

### Final Answer
Thus, the distance between the points \((\frac{3}{5}, 2)\) and \((- \frac{1}{5}, \frac{2}{5})\) is:

\[ d = \frac{4\sqrt{2}}{5} \]