Question

PQR is an equilateral triangle having perimeter equal to 45 metres. If ‘O’ is the centroid of the equilateral triangle PQR, then find the length of OP

• A. \(5\sqrt3 \) cm
• B. \(8\sqrt5 \) cm
• C. \(10\sqrt3 \) cm
• D. \(5\sqrt5 \) cm

Answer 1

The Correct Option is A

According to the question,
PQ = QR = PR = \(\frac{45}{3}\) = 15 cm
In triangle PQR, If PR = a, then MR = (\(\frac{a}{2}\)) (because ‘O’ is centroid of triangle therefore, PM will be median of the median and altitude of the triangle)
Using Pythagoras theorem in triangle PMR, we get
PM = \(\sqrt{a^2-(\frac{a}{2})^2 }\) = \(\sqrt{\frac{3a}{2}}\)
Therefore, PM = \(15\frac{\sqrt3}{2}\) cm
We know, centroid divide the median in the ratio 2 : 1
Therefore, OP = \(15\frac{\sqrt{3}}{2} × (\frac{2}{3})\) = \(5\sqrt3\) cm
So, the correct option is (A) : \(5\sqrt3\) cm.