8. The right-angled triangle shown has a base which is 4 times its height. Four such triangles are placed so that their hypotenuses form the boundary of a large square as shown.
What is the side length of the shaded square in the diagram?
A $2 x$
B $2 \sqrt{2} x$
C $3 x$
D $2 \sqrt{3} x$
E $\sqrt{15} x$
Solution
B The side length of the large square is $4 x$ and hence the area of this square is $16 x^{2}$. Each triangle has base $4 x$ and height $x$ and hence has area $\frac{1}{2}(4 x \times x)=2 x^{2}$. So the total area of these four triangles is $8 x^{2}$. Therefore the area of the shaded square is $16 x^{2}-8 x^{2}=8 x^{2}$. Therefore the side length of the shaded square is $\sqrt{8 x^{2}}=\sqrt{8} x=2 \sqrt{2} x$.