O and C are respectively the orthocentre and the circumcentre of an acute angled triangle $∆\mathrm{PQR}$. $\angle \mathrm{QCR}=138°$. A perpendicular PM is dropped from P on side QR. If $\angle \mathrm{PQR}=52°$, what is the degree measure of $\angle \mathrm{RPM}$?

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The correct option is 31.

Here is the step-by-step logical explanation and mathematical derivation:

Step 1: Relate the circumcentre angle to the vertex angle.
We are given that $C$ is the circumcentre of the acute-angled triangle $∆\mathrm{PQR}$, and the angle at the circumcentre is:
$\angle \mathrm{QCR}=138°$

According to the circumcentre theorem, the angle subtended by an arc at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle. Therefore, the angle at vertex $P$ is half of the circumcentre angle:
$\angle \mathrm{QPR}=\frac{1}{2}\angle \mathrm{QCR}$

Substituting the given value:
$\angle \mathrm{QPR}=\frac{138°}{2}=69°$

Step 2: Find the third angle of triangle PQR.
We are given that $\angle \mathrm{PQR}=52°$.
Using the angle sum property of a triangle for $∆\mathrm{PQR}$, we have:
$\angle \mathrm{QPR}+\angle \mathrm{PQR}+\angle \mathrm{PRQ}=180°$

Substituting the known values:
$69°+52°+\angle \mathrm{PRQ}=180°$

Simplifying the equation:
$121°+\angle \mathrm{PRQ}=180°$
$\angle \mathrm{PRQ}=180°-121°=59°$

Step 3: Calculate the measure of angle RPM.
A perpendicular $\mathrm{PM}$ is dropped from vertex $P$ onto the side $\mathrm{QR}$. This creates a right-angled triangle $∆\mathrm{PMR}$ at point $M$, which means:
$\angle \mathrm{PMR}=90°$

In the right-angled triangle $∆\mathrm{PMR}$, the sum of the interior angles is also $180°$:
$\angle \mathrm{RPM}+\angle \mathrm{PMR}+\angle \mathrm{PRM}=180°$

Note that $\angle \mathrm{PRM}$ is the same angle as $\angle \mathrm{PRQ}$ which is $59°$. Substituting the values:
$\angle \mathrm{RPM}+90°+59°=180°$

Simplifying the equation:
$\angle \mathrm{RPM}+149°=180°$
$\angle \mathrm{RPM}=180°-149°=31°$

Thus, the degree measure of $\angle \mathrm{RPM}$ is indeed 31.