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Prove that the area of an equilateral t...

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.

Text Solution

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Let `a` is the side of given triangle, then its diagonal will be,
`sqrt(a^2+a^2) = sqrt2a`
As, all equilateral triangles are similar, ratio of their ares will be squares of ratio of their sides.
So, ratio of area of given triangles,
`=a^2/(sqrt2a)^2 = a^2/(2a^2) = 1/2`
Thus, 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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Knowledge Check

  • Fill in the blanks . (i) In two similar triangles , ABC and PQR , if their corresponding altitudes AD and PS are in the ratio 4 : 9 , then ar(DeltaABC):ar(DeltaPQR)=overline((P)) . (ii) Area of an equilateral triangle described on the side of a square is overline((O)) the area of equilateral triangle described on its diagonal . (iii) The altitude of an equilateral triangle with side 'a' equals oberline((R ))

    A
    `{:(P,Q,R),(16:81,"Twice",(a)/(2)):}`
    B
    `{:(P,Q,R),(4 : 9,"Half",2a):}`
    C
    `{:(P,Q,R),(16 :81,"Half",(sqrt(3a))/(2)):}`
    D
    `{:(P,Q,R),(4 : 9,"Twice",(sqrt(3a))/(2)):}`

  • What is the area of an equilateral triangle with side 2cm?

    A
    `4sqrt(3)cm^(2)`
    B
    `3sqrt(3)cm^(2)`
    C
    `6cm^(2)`
    D
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