Question

In $\Delta ABC,D,E$ and $G$ are the mid-points of AB, AC and AE respectively. If $DE||BC,FG||DE,GH||AD,AG=4cm,BC=10cm$ and $DB=6cm$, then the perimeter of $\Delta FGH$ is equal to

• A. 10 cm
• B. 9.5 cm
• C. 9 cm
• D. 8.5 cm

Answer 1

The correct option is B 9.5 cm



In ∆ABC, D and E are mid-points of AB and AC respectively.
Also, $DE||BC$.
$\therefore$ By mid-point theorem, $DE=\frac{1}{2}BC=\frac{10}{2}=5cm$
Similarly, in $\Delta ADE,FG||DE$ and G is the mid-point of AE.
$\therefore$ By converse of mid-point theorem, F is the mid-point of AD and $FG=\frac{1}{2}DE=\frac{5}{2}=2.5cm$
Also, $GH||AD$
$\Rightarrow GH=\frac{1}{2}AD$ and H is the mid-point
$\Rightarrow GH=\frac{1}{2}\times BD$ ($\because AD=BD$)
$=\frac{1}{2}\times 6=3$ cm
Now, in $\Delta ADE,F$ and $H$ are mid-point of $AD$ and $DE$ respectively.
$\therefore$ By mid-point theorem,
$FH=\frac{1}{2}AE=AG=4$cm
$\therefore$ Perimeter of $\Delta FGH=FG+GH+FH$
$=2.5+3+4$
$=9.5$ cm
Hence, the correct answer is option (b).