Points M and N are on the sides PQ and QR respectively of a triangle PQR, right angled at Q. If PN = 9 cm, MR = 7 cm, and MN = 3 cm, then find the length of PR (in cm).

11

Let the measure of the side PM, MQ, QN, NR be d, a, b and c respectively.

From \(\Delta \)PQR, \( { (a\; + \;d)}^{2 } \) + \( { (b\; +\; c)}^{2 } \) = \( { PR}^{2 } \)

So, \(\Delta \)PQN, \(\Delta \)MQN & \(\Delta \)MQR are all right-angled triangles.

Now,

From \(\Delta \)PQN,

= \( { (a\; +\; d)}^{2 } \) + \( { b}^{2 } \) = \( { 9}^{2 } \)

= \( { (a\; +\; d)}^{2 } \) + \( { b}^{2 } \) = 81    ..(1)

From \(\Delta \)MQN,

\( { a}^{2 } \) + \( { b}^{2 } \) = \( { 3}^{2 } \)

= \( { a}^{2 } \) + \( { b}^{2 } \) = 9   ..(2)

From \(\Delta \)MQR,

= \( { a}^{2 } \) + \( {b\;+\;c}^{2 } \) = \( { 7}^{2 } \)

= \( { a}^{2 } \) + \( {b\;+\;c}^{2 } \) = 49   ..(3.)

(1) + (3)

= \( { (a\; +\; d)}^{2 } \) + \( { b}^{2 } \) + \( { a}^{2 } \) + \( {b\;+\;c}^{2 } \) = 81 + 49

= \( { (a\; +\; d)}^{2 } \) + \( { (b\; +\; c)}^{2 } \) + \( { b}^{2 } \) + \( { a}^{2 } \) = 130 

= \( { (a\; +\; d)}^{2 } \) + \( { (b\; +\; c)}^{2 } \) + 9 = 130

= \( { (a\; +\; d)}^{2 } \) + \( { (b\; +\; c)}^{2 } \) = 121

= \( { PR}^{2 } \) = 121

= PR = 11 cm

Therefore, PR is 11 cm.