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Q7   Prove that the area of an equilateral triangle described on one side of a square is equal
        to half the area of the equilateral triangle described on one of its diagonals.

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Let ABCD be a square of side units.

Therefore, diagonal = \sqrt{2}a

Triangles form on the side and diagonal are \triangle ABE and \triangleDEF, respectively.

Length of each side of triangle ABE = a units

Length of each side of triangle DEF = \sqrt{2}a units

Both the triangles are equilateral triangles with each angle of 60 \degree.

\triangle ABE\sim \triangle DBF      ( By AAA)

Using area theorem,

\frac{ar(\triangle ABC)}{ar(\triangle DBF)}=(\frac{a}{\sqrt{2}a})^2=\frac{1}{2}

Posted by

seema garhwal

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