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
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
Updated On: Apr 28, 2025
- \(5\sqrt3 \) cm
- \(8\sqrt5 \) cm
- \(10\sqrt3 \) cm
- \(5\sqrt5 \) cm
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The Correct Option is A
Solution and Explanation
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.
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