Let A B C D be a quadrilateral such that there exists a point E inside the quadrilateral satisfying A E=B E=C E= D E. Suppose \\angleD A B, \\angleA B C, \\angleB C D is an arithmetic progression. Then the median of the set \{\\angleD A B, \\angleA B C, \\angleB C D\} is

KVPY 2020-21 SA Exam Previous Papers

Question : 6 of 80

Marks: +1, -0

Let

A B C D

be a quadrilateral such that there exists a point

E

inside the quadrilateral satisfying

A E=B E=C E=

D E

. Suppose

\∠D A B, \∠A B C, \∠B C D

is an arithmetic progression. Then the median of the set

\{\∠D A B, \∠A B C

\∠B C D\}

${\π}/{6}$
${\π}/{4}$
${\π}/{3}$
${\π}/{2}$

Validate

Solution:

∵ ∠DAB, ∠ABC and ∠BCD are in A.P.
∴ Let ∠DAB = θ – α, ∠ABC = θ and ∠BCD = θ + α
∴ Median of ∠DAB, ∠ABC and ∠BCD = θ
From point E all the vertices are at equal distance.
∴ quadrilateral is cyclic.
and ∠ADC = 2π – (θ – α + θ + θ + α) = 2π –3θ
and ∠ADC + ∠ABC = π
⇒ 2π – 3θ + θ = π

⇒θ=π/2

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