Question

Let $ABCD$ is a square with sides of unit length .Points E and F are taken on sides $AB$ and $AD$ respectively so that $AE=AF$ .Let P be a point inside the $\square ABCD$.
Let a line passing through point $A$ divides the square $\square ABCD$ into two parts so that area of one portion is double the other, then the length of portion of line inside the square is

• A. $\frac{\sqrt{10}}{3}$
• B. $\frac{\sqrt{13}}{3}$
• C. $\frac{\sqrt{11}}{3}$
• D. $\frac{2}{\sqrt{3}}$

Answer 1

Answer: (B)

$\frac{\sqrt{13}}{3}$

ABCD is square with unit sides

A divides $\square$ABCD in two parts such that one is double of other, from figure.

$A\left(\square ABCD\right)=1$ $uni{t}^2$ ...........$\left(1\right)$

& line Ax divides $\square ABCD$ such that,

$A\left(\square ABCX\right)=2A\left(\Delta ADX\right)$ ...........$\left(2\right)$

From $\left(1\right)$ & $\left(2\right)$

$A(\Delta ADX=\frac{1}{3}$ units

Also $A\left(\Delta ADX\right)=\frac{1}{2}\times DX\times AD$

$\frac{1}{3}=\frac{1}{2}\times DX\times 1$

$DX=\frac{2}{3}$

$\therefore$ For AX, using pythagorus theorem

$AX=\sqrt{A{D}^2+D{X}^2}$

$=\sqrt{\frac{4}{9}+1}$

$AX=\frac{\sqrt{13}}{3}$

Length of line inside square.