how to construct a triangle pqr with givenin which three sides are in the ratio 1:3:5 and its perimeter is 13.5 cm.

Please check your question because for the given ratio the triangle is not possible.

Here the sides of triangle are in the ratio 1:3:5.
So, Suppose the sides of triangle are x, 3x and 5x.
So, perimeter of triangle = x + 3x + 5x = 9x 
And the given perimeter of triangle = 13.5 cm
So, we have;
9x = 13.5
$\Rightarrow x = \frac{13.5}{9} = 1.5$
So, the sides of triangle are 1.5, $3\times 1.5$, $5\times 1.5$ i.e. 1.5 cm, 4.5 cm and 7.5 cm
Now for a triangle the sum of two sides should be greater than third side.
But, sum of two sides, 1.5 + 4.5 = 6 < 7.5
Therefore, the triangle is not possible for the given ratio.