Two medians QT and RS of the triangle PQR intersect at G at right angles. If QT = 18 cm and RS = 27 cm. then the length of QS is:

15 cm

The two medians intersect at G. Hence G is the centroid of the triangle and the centroid divides the median in the ratio 2 : 1.

Therefore, QG : GT = 2 : 1 = QG = 18 x \(\frac{2}{3}\) = 12 cm

Also, RG : GS = 2 : 1 = GS = 27 x \(\frac{2}{3}\) = 9 cm.

Therefore, In \(\Delta \)GQS, \(\angle\)QGS = 90  (as medians intersect at 90 degrees.)

Using pythagoras theorem

\( {QG }^{2 } \) + \( {GS }^{2 } \) = \( {QS }^{2 } \)

= \( {12 }^{2 } \) + \( {9 }^{2 } \) = \( {QS }^{2 } \)

= \( {QS }^{2 } \) = 225

= QS = 225

= QS = \(\sqrt {225 }\) = 15 cm.

Therefore, the length of QS is 15 cm.