CAT 2022 Slot 2 Paper

Question : 66 of 66

Marks: +1, -0

The length of each side of an equilateral triangle $\ABC$ is 3 cm. Let $\D$ be a point on $\BC$ such that the area of triangle $\ADC$ is half thearea of triangle $\ABD$ . Then the length of $\AD$ , in cm, is

$√8$
$√6$
$√7$
$√$

Validate

Solution:

Area of triangle

{\ABD}

is twice the area of triangle

\ACD

∠ \ADB=\θ

{1}/{2}(\AD)(\BD) \sin \θ=2 ({1}/{2}(\AD(\CD) \sin (180-\θ)))

\BD=2 \CD

Therefore,

{\BD}=2

and

\CD=1

∠ \ABC=∠ \ACB=60°

Applying cosine rule in triangle

\ADC

, we get

\cos ∠ \ACD={\AC^2+\CD^2- \AD^2}/{2(\AC)(\CD)}

{1}/{2}={9+1-\AD^2}/{6}

\AD^2=7

\AD={√{7}}

The answer is option C.

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