KVPY 2020-21 SA Exam Previous Papers

Question : 62 of 80

Marks: +1, -0

. Let ABC be a triangle such that AB = 4, BC = 5 and CA = 6. Choose points D, E, F on AB, BC, CA respectively, such that AD = 2, BE = 3, CF = 4. Then

{\areaΔDEF}/{\areaΔABC}

Solution:

Here a = 5, b = 6 and c = 4
Area of

\ΔA D F={1}/{2} \.2 \.2 \sin A=2 \sin A

Area of

\ΔB D F={1}/{2} \. 2 \.3 \sin B=3 \sin B

and area of

\ΔC E F={1}/{2} \.2 \.4 \sin C=4 \sin C

and area of

\ΔA B C=\Δ

\∴{\text " area of " \ΔD E F}/{\text "area of " \ΔA B C}

={\Δ-(2 \sin A+3 \sin B+4 \sin C)}/{\Δ}

=1-{1}/{\Δ}(2 \.{2 \Δ}/{b c}+3 \.{2 \Δ}/{a c}+4 \.{2 \Δ}/{a b})

=1-{2(2 a+3 b+4 c)}/{a b c}

=1-{2(10+18+16)}/{5 \.6 \.4}={16}/{60}={4}/{15}

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