# Rectangle

In rectangle with sides, 6 and 3 mark the diagonal. What is the probability that a randomly selected point within the rectangle is closer to the diagonal than to any side of the rectangle?

Correct result:

p =  42.7 %

#### Solution:

$a=6 \ \\ b=3 \ \\ c=\sqrt{ a^2+b^2 }=\sqrt{ 6^2+3^2 } \doteq 3 \ \sqrt{ 5 } \doteq 6.7082 \ \\ \ \\ S_{1}=\dfrac{ a \cdot \ b }{ 2 }=\dfrac{ 6 \cdot \ 3 }{ 2 }=9 \ \\ \ \\ r=\dfrac{ 2 \cdot \ S_{1} }{ a+b+c }=\dfrac{ 2 \cdot \ 9 }{ 6+3+6.7082 } \doteq 1.1459 \ \\ S_{2}=\dfrac{ r \cdot \ c }{ 2 }=\dfrac{ 1.1459 \cdot \ 6.7082 }{ 2 } \doteq 3.8435 \ \\ \ \\ p=100 \cdot \ \dfrac{ S_{2} }{ S_{1} }=100 \cdot \ \dfrac{ 3.8435 }{ 9 }=42.7 \%$

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