PQRS is a quadrilateral whose diagonals are perpendicular to each other. If PQ = 16cm, QR = 12cm and RS = 20cm, find PS.

To find the length of the diagonal PS in the quadrilateral PQRS, we can use the properties of a quadrilateral with perpendicular diagonals.

Step 1: Since the diagonals are perpendicular and bisect each other, quadrilateral PQRS is a rhombus.

Step 2: In a rhombus, the diagonals bisect each other at 90 degrees and they also divide the rhombus into four congruent right-angled triangles.

Step 3: We know that PQ = 16cm, QR = 12cm and RS = 20cm. Since PQRS is a rhombus, PQ = RS = 16cm and QR = PS = 12cm.

Step 4: Now, we can use the Pythagorean theorem to find the length of the diagonal PS. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Step 5: Let's consider one of the right-angled triangles formed by the diagonals. The hypotenuse is PS (which we are trying to find), one side is half of PQ which is 16/2 = 8cm, and the other side is half of RS which is 20/2 = 10cm.

Step 6: Applying the Pythagorean theorem, we get PS^2 = (8cm)^2 + (10cm)^2 = 64cm^2 + 100cm^2 = 164cm^2.

Step 7: Therefore, the length of the diagonal PS = sqrt(164cm^2) = 12.81cm (rounded to two decimal places).

So, the length of the diagonal PS in the quadrilateral PQRS is approximately 12.81cm.