Q.332. Plane geometry

Question 332.
A square with a side length "x" is inscribed in an equilateral triangle with a side length "t". If t = kx.
find k to the nearest thousandths place.

Answer 332.
Refer to the figure in the link as under.

ABCD is a square inscribed in triangle PQR.
PM is perpendicular to the base QR.

AD = x and
QD = QM - MD = t/2 - x/2

tan ∠PQM = AD/QD
=> tan60° = x / (t/2 - x/2)
=> √3 = 2x / (t - x)
=> √3 (t - x) = 2x
=> √3 t = (2 + √3) x
=> k = t/x = (2 + √3) / √3 ≈ 2.155.
Link to YA!