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Equal Angles Subtend Equal Sides

In the figure...

Question

In the figure $Δ$ ABC is an equilateral triangle with each side of length $10$ cm and Q is a point of the interior of the triangle. If P , Q and R are mid points of AO,BO, and CO respectively , then find the perimeter of $Δ$ PQR.

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Solution

$Q$ cannot be interior and mid point at the same time
By basic proportionality theorm
$P Q$ is $\frac{1}{2} \times A B = 5$
This happens in all the $3$ cases
So Perimeter is $15$

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