Equilateral triangle and square

Q.6: There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral △. What is the ratio of the area of the square to that of the equilateral triangle?

1. 12 : 12 + 7$\sqrt {3}$
2. 24 : 24 + 7$\sqrt {3}$
3. 18 : 12 + 15$\sqrt {3}$
4. 6 : 6 + 5$\sqrt {3}$

Explanatory Answer

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Detailed Solution

APQ is an equilateral △. As PQ is parallel to BC.
Let side of the square be ‘a’
AP = a = AQ
△QRC has angles 30 – 60 – 90.
${{QR \over QC} = {{\sqrt {3}} \over 2}}$
QC = QR x ${2 \over {\sqrt {3}}}$

AC = AQ + QC
a + ${2a \over {\sqrt {3}}}$

Area of equilateral △

${{{\sqrt {3}} \over 4}{AC^2}}$

${{{\sqrt {3}} \over 4} {({a} + {2a \over {\sqrt {3}}})^2}}$

${{{\sqrt {3}} \over 4} {({{\sqrt {3}}a + 2a \over {\sqrt {3}}})^2}}$

${{{\sqrt {3}} \over 4} {*} {{1 \over 3}a^2} ({4} + {3} + {4}{\sqrt {3}})}$

${{{\sqrt {3}}a^2{({7} + 4{\sqrt 3})}} \over 12}$

${{a^2{({12} + 7{\sqrt 3})}} \over 12}$

Ratio of area of square to that of equilateral △ is = ${{a^2} \over {{a^2{({12} + 7{\sqrt 3})}} \over 12}}$ = ${12 \over {{12} + 7{\sqrt 3}}}$
Answer choice (a)

Correct Answer: 12 : 12 + 7$\sqrt {3}$

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