CAT 2020 Exam Shift 2 Solved Paper

Question : 74 of 76

Marks: +1, -0

From an interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the lengths of the three perpendiculars is s. Then the area of the triangle is

$\;{√{3} s^{2}}/{2}$
$\;{s^{2}}/{√{3}}$
$\;{2 s^{2}}/{√{3}}$
$\;{s^{2}}/{2 √{3}}$

Validate

Solution:

{PD}+ {PE}+ {PF}= {s}

Area of

=\;{1}/{2} × A B × P E+\;{1}/{2} × B C × P D+\;{1}/{2} × A C × P F

A B=B C=C A

, we've

=\;{1}/{2} × A B(P D+P E+P F)=\;{1}/{2} A B × s-(1)

Now

\;{√{3}}/{4} A B^{2}=\;{1}/{2} A B × s

⇒ A B=\;{2}/{√{3}} s

Required value

=\;{1}/{2} × \;{2}/{√{3}} × s^{2}=\;{s^{2}}/{√{3}}

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