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31 Mar 2019, 22:55
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
51% (03:03) correct
49%
(02:23)
wrong
based on 69
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A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
A. \(\frac{\sqrt{2} - 1}{2}\)
B. 1/4
C. \(\frac{2 - \sqrt{2}}{2}\)
D. \(\frac{\sqrt{2}}{4}\)
E. \(2 - \sqrt{2}\)
Attachment:
7274a387184a2ce0dc35e8ecaac8b5da8b052e4c.png [ 2.88 KiB | Viewed 4085 times ]
02 Apr 2019, 08:07
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IMO A
Let the side of octagon be r
exterior angle of octagon = 360/8 = 45
So interior angle = 180 -45 = 135
So all the triangles will have 45-90-45 combinations
c sides of triangles = r/\sqrt{2}
Calculate the total area of octagon and the area of central square
Probability = area of square /area of octagon = r^2/(r^2+r^2+2\sqrt{2}+r^2)
= 1/(2+2\sqrt{2})
take 2 common and multiply and divide by \sqrt{2} -1
we get option A as answer
Award kudos if helpful
Let the side of octagon be r
exterior angle of octagon = 360/8 = 45
So interior angle = 180 -45 = 135
So all the triangles will have 45-90-45 combinations
c sides of triangles = r/\sqrt{2}
Calculate the total area of octagon and the area of central square
Probability = area of square /area of octagon = r^2/(r^2+r^2+2\sqrt{2}+r^2)
= 1/(2+2\sqrt{2})
take 2 common and multiply and divide by \sqrt{2} -1
we get option A as answer
Award kudos if helpful
18 Sep 2019, 08:29
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Let the side of octagon be r
exterior angle of octagon = 360/8 = 45
So interior angle = 180 -45 = 135
So all the triangles will have 45-90-45 combinations
c sides of triangles = \(r/\sqrt{2}\)
Calculate the total area of octagon and the area of central square
Probability = area of square /area of octagon = \(r^2/(r^2+r^2+2\sqrt{2}+r^2)\)
= \(1/(2+2\sqrt{2})\)
take 2 common and multiply and divide by \(\sqrt{2} -1\)
we get option A as answer
P.S: I just formatted the post of m1033512. My solution is in next comment
exterior angle of octagon = 360/8 = 45
So interior angle = 180 -45 = 135
So all the triangles will have 45-90-45 combinations
c sides of triangles = \(r/\sqrt{2}\)
Calculate the total area of octagon and the area of central square
Probability = area of square /area of octagon = \(r^2/(r^2+r^2+2\sqrt{2}+r^2)\)
= \(1/(2+2\sqrt{2})\)
take 2 common and multiply and divide by \(\sqrt{2} -1\)
we get option A as answer
P.S: I just formatted the post of m1033512. My solution is in next comment
