The above shows two squares $ABCD$ and $EFGH$ with side lengths 10 and 6, respectively. Given that $A,E,F$ and $B$ lies on a straight line. And the straight line $AE$ has a distance of 1.

If $\vec{AI} = \beta \vec{EI}$ , what is the value of $\beta$ ?

$\frac{3}{5}$
$\frac{1}{2}$
$\frac{5}{3}$
$\frac{2}{3}$

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Triangles $ADI and\ EHI$ are similar triangles by A.A.A.

$\angle I= \angle I$ Reflexive angle.

$\angle D=\angle H$ Corresponding angles of parallel lines

$\angle A=\angle E$ = $90 ^ {deg}$ Corresponding angles of parallel lines.

By taking the ratio of proptionality $\frac{AI}{EI}$ = $\frac{10}{6}$ = $\frac{5}{3}$

Keeping in mind two perpendiculars to the same line are parallel.