Question

if the median if a n equilateral triangle is x , find the area of triangle

Answer 1

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$\underline{SOLUTION}$ ➡

$We \: know \: \: that ,\: \\ \\ In \: an \: equilateral \: triangle \: \: the \: median \: is \: \\ also \: the \: altitude \: (i.e \: \: height) \: of \: the \: triangle \: . \\ \\ Given, \\ \\ Median \: of \: the \: equilateral \: triangle \: = x$

$In \: the \: above \: figure, \\ \\ Let ,\\ \\ ABC\: \: be \: the \: equilateral \: triangle. \\ \\ AD \: \: is \: the \: \: median \: as \: well \: as \: the \: altitude \: . \\ \\ \\ So ,\\ \\ AD \: = x \\ \\ Let ,\\ \\ Each \: sides \: are \: = a \\ \\ Therefore ,\\ \\ AB = BC = CA = a \\ \\ So ,\\ \\ BD = DC = \frac{a}{2}$

$Now,\\ \\ In \: the \: right \: angled \: triangle \: ADB ,\\ \\ AB{}^{2} = BD {}^{2} + AD {}^{2} \\ \\ = > a {}^{2} = ( \frac{a}{2} ) {}^{2} + x {}^{2} \\ \\ = > a {}^{2} = \frac{ a{}^{2} }{4} + x {}^{2} \\ \\ = > a {}^{2} - \frac{a {}^{2} }{4} = x {}^{2} \\ \\ = > \frac{3a {}^{2} }{4} = x {}^{2} \\ \\ = > 3a {}^{2} = 4x {}^{2} \\ \\ = > a {}^{2} = \frac{4x {}^{2} }{3} \\ \\ = > a = \sqrt{ \frac{4 x{}^{2} }{3} } \\ \\ = > a = \frac{2x}{ \sqrt{3} }$

$So ,\\ \\ Area \: of \: the \: equilateral \: \\ triangle, \: ABC \: = \frac{ \sqrt{3} }{4} a {}^{2} \\ \\ = \frac{ \sqrt{3} }{4} \times ( \frac{2x}{ \sqrt{3} } ) {}^{2} \\ \\ = \frac{ \sqrt{3} }{4} \times \frac{4x {}^{2} }{3} \\ \\ = \frac{x {}^{2} \sqrt{3} }{3} \\ \\ = \frac{x {}^{2} \times \sqrt{3} }{ \sqrt{3} \times \sqrt{3} } \\ \\ = \frac{x {}^{2} }{ \sqrt{3} }$

$\underline{ANSWER}$ ➡

$Area \: = \frac{x {}^{2} }{ \sqrt{3} }$

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Answer 2

Median of the triangle = x

We know that in an equilateral triangle, the median is also the altitude (that is, height) of the triangle.
Let the side of the triangle be = y divided by two.
In right angled triangle ADB,
AB square =BD square +AD square
Which implies, y square = y upon 4 square + x square. Which Implies, x square = y square minus y divided by 4 square
Which implies, x square =3y upon 4 square
Which implies, x = root 3y upon 2
Which implies, y=2x upon root 3
Therefore, area of triangle ABC =1 upon 2 into 2x upon root 3 into x= x square upon root 3cm square.

Hope this answer will help you.