In an equilateral triangle PQR, PA is the altitude. O is the orthocentre and PO=12 the find the value of PA.

PQR is a equilateral triangle PA is the altitude, O is the orthocentre and PO \$= 12\$

In an equilateral traingle all the points such as orthocentre, centroid, circumcenter coincide.

So O is the centroid and also the centroid.

Centroid divides the median in the ratio \$2:1\$.

\begin{align} \require{cancel} &&\therefore PO:OA &= 2:1 \\ \end{align}

If PO = 12, then OA is

\begin{align} \require{cancel} && \frac{PO}{OA} &= \frac{2}{1} \\ \Rightarrow && \frac{12}{OA} &= \frac{2}{1} \\ \Rightarrow && \frac{\cancel{12} ^{6} }{OA} &= \frac{\cancel{2} ^{1}}{1} \\ \Rightarrow && \frac{6}{OA} &= \frac{1}{1} \\ \Rightarrow && OA &= 1 \\ \end{align}

Now, consider

\begin{align} \require{cancel} PA &= PO + PA \\ PA &= 12 + 6 \\ PA &= 18 \\ \end{align}

Answer

The value of PA is 18.