Given: △pqr, m∠r = 90°, mp = 18 m∠pqr = 75°, m∠mqr = 60° find: rq

Correct answer is RQ=9

Solution:-

Given that in a ΔPQR, m∠R=90°, MP=18, m∠PQR=75° and m∠MQR=60°.

Since m∠MQR<m∠PQR, M should lie on PR only, Please refer the diagram attached for reference.

From the diagram∠PQM=∠PQR-∠MQR=75°-16°=15°

And in ΔPQR,∠RPQ=180-(remaining 2 angles in triangle)

                                =180-(90+75)=15°

So, in ΔPQM, ∠MPQ=∠PQM hence the opposite sides must be equal that is

    MP=MQ but given MP=18

Hence MQ=18

In ΔMRQ, cos(∠MQR)=

                cos(60°)=

                RQ= 18cos(60°)=18X0.5 =9

Hence RQ=9

RQ = 9

Step-by-step explanation:

In the given triangle,

∠R = 90° , ∠PQR = 75° , ∠MQR = 60° and MP = 18

From the diagram

                       ∠RPQ = ∠ PRQ - ∠PQR = 90° - 75° = 15°·············(1)

In ΔMPQ

               ∠MQP = ∠PQR - ∠MQR = 75° - 60° = 15°·················(2)

from (1) and (2)

                          ∠RPQ = ∠MQP = 15°

So, ΔPMQ is an isosceles triangle.

Hence,

            MP = MQ = 18

Now, in ΔMRQ

              cos(60°) =

                 =

                RQ = 0.5 × 18

                 RQ = 9

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