Label the vertices A:(2,9);B:(3,0) and C(1,3).

To find the perimeter of the triangle we find the lengths of each of the sides. To find the length, use the distance formula: between points `(x_1,y_1),(x_2,y_2) ` the distance `d ` is given by `d=sqrt((x_2-x_1)^2+(y_2-y_1)^2) ` . Note that the distance formula is just the Pythagorean theorem.

`AB=sqrt(1^2+9^2)=sqrt(82) `

`BC=sqrt(2^2+3^2)=sqrt(13) `

`AC=sqrt(1^2+6^2)=sqrt(37) `

Then the perimeter is ` p=sqrt(82)+sqrt(13)+sqrt(37) `

The distance between two points (x1, y1) and (x2, y2) is given by:

**sqrt[(x2 - x1)^2 + (y2 - y1)^2]**

So, the distance between (2,9) and (3,0) =

sqrt[(3 - 2)^2 + (0 - 9)^2] = sqrt(1 + 81) = sqrt(82)

Similarly, the distance between (3,0) and (1,3) =

sqrt[(1 - 3)^2 + (3 - 0)^2] = sqrt(4 + 9) = sqrt(13)

The distance between (1,3) and (2,9) =

sqrt[(2 - 1)^2 + (9 - 3)^2] = sqrt(1 + 36) = sqrt(37)

These are actually the lengths of the sides of the triangle.

Therefore, the **perimeter** of the triangle = sqrt(82) + sqrt(13) + sqrt(37)

= 9.055 + 3.606 + 6.083 = **18.744**