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?

Result

p =  42.7 %

Solution:

a=6 b=3 c=a2+b2=62+32=3 56.7082  S1=a b2=6 32=9  r=2 S1a+b+c=2 96+3+6.70821.1459 S2=r c2=1.1459 6.708223.8435  p=100 S2S1=100 3.8435942.7051=42.7%a = 6 \ \\ b = 3 \ \\ c = \sqrt{ a^2+b^2 } = \sqrt{ 6^2+3^2 } = 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 } \doteq 42.7051 = 42.7 \%



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Tips to related online calculators
Our percentage calculator will help you quickly calculate various typical tasks with percentages.
Pythagorean theorem is the base for the right triangle calculator.
See also our trigonometric triangle calculator.
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