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A dart is thrown and hits the square target in the diagram...


The target has a circle inscribed in it, and within that circle, is inscribed another square. If the dart is equally likely to hit any point on the target, what is the probability that the dart will land on the shaded area?

Respuesta :

r - radius of a circle
d - diagonal of a square inscribed in a circle
a - side of a square inscribed in a circle
shaded area - area of a square inscribed in a circle

[tex]A_{circle}=r^2\pi \\d=2r=a \sqrt{2} \\ \\a= \frac{2r}{ \sqrt{2} } =r \sqrt{2} \\ \\A_{square}=a^2=(r \sqrt{2} )^2=2r^2 \\ \\P= \frac{A_{square}}{A_{circle}} = \frac{2r^2}{r^2\pi}= \frac{2}{\pi}\approx0.64\times 100\%=64\% [/tex]