Question

Question 7
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.

Answer 1

Given, ABCD is a square whose one diagonal is AC.
$\Delta APC$ and $\Delta BQC$ are two equilateral triangles described on the diagonal AC and side BC of the square ABCD.
To Prove that $area\left(\Delta BQC\right)=\frac{1}{2}area\left(\Delta APC\right)$
Proof
$\Delta APC$ and $\Delta BQC$ are both equilateral triangles (Given)
$\therefore \Delta APC\sim \Delta BQC$ [AAA similarity criterion]
$\therefore \frac{area\left(\Delta APC\right)}{area\left(\Delta BQC\right)}=\frac{A{C}^2}{B{C}^2}$
$\Rightarrow {\left(\frac{\sqrt{2}BC}{BC}\right)}^2=\frac{2B{C}^2}{B{C}^2}=2[Since,Diagonal=\sqrt{2}side=\sqrt{2}BC]$
$\Rightarrow area\left(\Delta APC\right)=2\times area\left(\Delta BQC\right)$
$\Rightarrow area\left(\Delta BQC\right)=\frac{1}{2}area\left(\Delta APC\right)$