Question

O is a point in the interior of a square ABCD such that OAB is an equilateral triangle. Show that triangle OCD is an isoceles triangle.

Answer 1

Given :  O is a point in the interior of a square ABCD such that OAB is an equilateral triangle

To Find : Show that triangle OCD is an isoceles triangle.

Solution:

ABCD is a square

Hence

AB = BC = CD = AD

∠A = ∠B  = ∠C = ∠D 90°

OAB is an equilateral triangle

Hence OA = OB = AB

& ∠OAB = ∠OBA  = ∠AOB = 60°

∠DAO = ∠A - ∠OAB = 90° - 60° = 30°

∠CBO = ∠B - ∠OBA = 90° - 60° = 30°

Compare Δ DAO & CBO

AD = BC    ( Sides of square )

∠DAO  = ∠CBO

OA = OB    ( side of Equilateral traingle )

=> Δ DAO ≅ CBO  ( SAS criteria )

=> OD = OC

in ΔOCD

OD = OC

hence isosceles triangle

QED

Hence proved triangle OCD is an isoceles triangle.

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Answer 2

Answer:

we know that angle OAB is an equilateral triangle

∆OAB =angle o p a barabar a 60 degree from the figure that ABCD is a square so we get ∆ A=B=C=D 90 degree

order to find the value of DAO

angle A =DOA+OAB

90 =DOA +60

DOA=90-60

30

OAB=OBA

OA=OB

OAD=∆DBC

there it proved that angle OCD is an isosceles triangle

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