Geometry

Find the side length of the square PQRS.

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The height of the triangle is h.

$A=\frac{1}{2}ah\\ h=\frac{2A}{a}=\frac{2\cdot 30cm^2}{6cm}=\color{blue}10cm$

$Prerequisite\ for\ a\ square\ on\ \overline{BC}.\\ 90^o\geq \ ∠ABC\ \geq arctan(\frac{10}{6})\\ $

The side length of the square is x.

$(10-x):10=x:6\\ 60-6x=10x\ |\ +6x\\ 60=16x\ |\ :16\\ \frac{60}{16}=x\\ \color{blue}x=3.75$

The side length of the square PQRS is 3.75cm.

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