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

O is a point in the interior of a square ABCD such that OAB is an equilateral triangle. Show that \triangleOCD is an isosceles triangle.

Answers (1)

ABCD is square. O is any interior point of the square and OAB is an equilateral triangle.

To prove: \triangleOCD is an isosceles triangle.

Proof : \angleAOB = \angleOAB = \angleOBA = 60°

 And,   DA = AB = OB                       (\because \triangleOAB is equilateral)

Now,   \angleDCB = \angleBAD = \angleABD = \angleCDA = 90°

DA = AB = CB = CD              (\because ABCD is square)

Let, \angleOAB = \angleOBA = 60°                ……(1)

\angleBAD = \angleABD = 90°                        …….(2)

Then subtract eqn. (1) from (2), we get

\angleBAD – \angleOAB =\angleABD – \angleOBA = 90° – 60° = 30°

\angleDAO = \angleCBO = 30° …….(3)

In \triangleAOD and \triangleBOC

AO = BO                     (OAB is an equilateral triangle)

\angleDAO = \angleCBO         (from 3)

AD = BC                     (ABCD is square)

\therefore\triangleAOD \cong \triangleBOC      (by SAS criterion of congruency)

\Rightarrow DO = OC                (by CPCT)

\therefore\triangleOCD is an isosceles triangle.

Hence proved

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