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

Answer: (A)

$5\sqrt3$ cm

Answer 2

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.