Triangle pqr is an equilateral triangle with each side as 6 centimetres and o is a point in the interior of the triangle if p q and R are respectively the mid points of xo yo and zo then find the perimeter of triangle xyz

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Please find below the solution to the asked query:
$$\begin{gathered} \mathrm{p} , \mathrm{q} \mathrm{and} \mathrm{r} \mathrm{are} \mathrm{the} \mathrm{mid} \mathrm{points} \mathrm{of} \mathrm{XO} , \mathrm{OY} \mathrm{and} \mathrm{OZ} \\ \mathrm{In} \mathrm{triangle} \mathrm{OZX} , \mathrm{p},\mathrm{r} \mathrm{is} \mathrm{mid} \mathrm{point} \mathrm{of} \mathrm{OX} \mathrm{and} \mathrm{OZ} \mathrm{so} \\ \mathrm{According} \mathrm{to} \mathrm{the} \mathrm{theorem} \mathrm{if} \mathrm{in} \mathrm{a} \mathrm{triangle} \mathrm{a} \mathrm{line} \mathrm{join} \mathrm{the} \mathrm{mid} \mathrm{point} \mathrm{of} \mathrm{two} \mathrm{line} \mathrm{is} \mathrm{parallel} \mathrm{to} \mathrm{the} \mathrm{third} \mathrm{line} \\ \mathrm{So} \mathrm{triangle} \mathrm{Orp} \mathrm{and} \mathrm{OXZ} \mathrm{are} \mathrm{similar} \mathrm{triangles} \\ \mathrm{Hence} \mathrm{the} \mathrm{ration} \mathrm{of} \mathrm{their} \mathrm{sides} \mathrm{also} \mathrm{same} \\ \Rightarrow \frac{\mathrm{Pr}}{\mathrm{XZ}}=\frac{\mathrm{OP}}{\mathrm{OX}} \\ \Rightarrow \frac{\mathrm{Pr}}{\mathrm{XZ}}=\frac{\mathrm{OP}}{2\mathrm{OP}}=\frac{1}{2} \\ \Rightarrow 2\mathrm{pr}=\mathrm{XZ} \\ \Rightarrow 2\times 6=\mathrm{XZ} \\ \Rightarrow \mathrm{XZ}=12 \\ \mathrm{Similarly} \mathrm{YZ}=\mathrm{YX}=12 \mathrm{cm} \\ \mathrm{So} \mathrm{perimeter} \mathrm{of} \mathrm{triangle} \mathrm{XYZ}=\mathrm{sum} \mathrm{of} \mathrm{their} \mathrm{sides} \\ \mathrm{Perimeter}=12+12+12=36 \mathrm{cm} \left(\mathrm{Answer}\right) \end{gathered}$$



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