Question

In the given figure, ABCD is a rectangle where E, F, G and H are the mid-points of AB, BC, CD and DA. If AB = 24 cm and BC = 18 cm, then the semi-perimeter of EFGH is equal to

• A. 42 cm
• B. 60 cm
• C. 45 cm
• D. 30 cm

Answer 1

The correct option is D 30 cm


Given that ABCD is a rectangle.
$\therefore AB=DC=24cm$
Also, E and G are mid-points of AB and CD respectively.
$\Rightarrow AE=EB=GC=GD=\frac{24}{2}=12cm$
Now, $BC=DA=18cm$
And, F and H are mid-points of BC and AD respectively.
$\Rightarrow BF=FC=DH=AH=\frac{18}{2}=9cm$
Also, $\angle HAE=\angle EBF=\angle FCG=\angle GDH=90°$ ($\because ABCD$ is rectangle)
In $\Delta AHE$, by Pythagoras theorem,
$(HE{)}^2=(AH{)}^2+(AE{)}^2$
$\Rightarrow (HE{)}^2=(9{)}^2+(12{)}^2=81+144$
$\Rightarrow HE=\sqrt{225}=15cm$
Similarly, $EF=FG=GH=15cm$
$\therefore$ Semi-perimeter of EFGH $=\frac{1}{2}(EF+FG+GH+HE)$
$=\frac{1}{2}\times 4\times 15$
$=30cm$
Hence, the correct answer is option (d).