MCQ
In $\triangle A B C$ points $P$ and $Q$ trisect side $A B$ points $T$ and $U$ trisect side $A C$ and points $R$ and $S$ trisect side $B C$. Then perimeter of hexagon $PQRSTU$ is how many times of the perimeter of $\triangle A B C$ ?
  • A
    $\frac{1}{3}\text{times}$
  • $\frac{2}{3}\text{times}$
  • C
    $\frac{1}{6}\text{times}$
  • D
    $\frac{1}{2}\text{times}$

Answer

Correct option: B.
$\frac{2}{3}\text{times}$

Let $AB$ be $x$
$\therefore AQ = QP = BP  = \frac{\text{x}}{3}$
Let $BC$ be $y$
$\therefore BR = RS = SC  = \frac{\text{y}}{3}$
Let $AC = z$
$AT = TU = UC = \frac{\text{z}}{3}$
Opposite sides of Hexagon are equal
$\therefore$ Perimeter of Hexagon $= PQ + QT + TU + US + RS + PR$
$ = \Big(\frac{\text{x}}{3} + \frac{\text{y}}{3} + \frac{\text{z}}{3}\Big) \times{2}$
$\therefore\frac{2}{3}$ Perimeter of hexagon is $\frac{2}{3}$ times the perimeter of $△ABC.$

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