In a triangle PQR, the lengths of sides PQ and PR are 18 cm and 24 cm, respectively. Point S lies on side QR such that PS is a median of the triangle. If the perimeter of triangle PQR is 72 cm, determine the length of PS.

The correct option is 15 cm.

Step 1: Calculate the length of side QR
We are given the following side lengths for triangle PQR:
PQ = 18 cm
PR = 24 cm
The perimeter of triangle PQR is given as 72 cm. Since perimeter is the total boundary length of the triangle:
Perimeter = PQ + PR + QR
72 = 18 + 24 + QR
72 = 42 + QR
QR = 72 - 42 = 30 cm

Step 2: Determine the length of segment QS
Point S lies on side QR such that PS is a median of triangle PQR. A median divides the side it intersects into two equal segments. Therefore, S is the midpoint of QR:
QS = SR = QR / 2
QS = 30 / 2 = 15 cm

Step 3: Apply Apollonius's Theorem to find PS
Apollonius's Theorem relates the lengths of the sides of a triangle to the length of its median. For triangle PQR with median PS to side QR:

${\mathrm{PQ}}^2+{\mathrm{PR}}^2=2({\mathrm{PS}}^2+{\mathrm{QS}}^2)$

Substituting the known values into the equation:

${18}^2+{24}^2=2({\mathrm{PS}}^2+{15}^2)$

$324+576=2({\mathrm{PS}}^2+225)$

$900=2({\mathrm{PS}}^2+225)$

Divide both sides by 2:

$450={\mathrm{PS}}^2+225$

Subtract 225 from both sides:

${\mathrm{PS}}^2=450-225$

${\mathrm{PS}}^2=225$

Taking the square root on both sides:

$\mathrm{PS}=\sqrt{225}=15$ cm

Thus, the length of the median PS is 15 cm.