Octagon problem

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artstudent: Senior | Next Rank: 100 Posts; Posts: 62; Joined: Mon Mar 28, 2011 9:35 pm; Thanked: 4 times

by artstudent » Sun Jul 10, 2011 12:20 am

Can someone please explain this?

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Source: Beat The GMAT — Problem Solving | Sun Jul 10, 2011 12:20 am

pemdas: Legendary Member; Posts: 1084; Joined: Fri Apr 15, 2011 2:33 pmFollowed by: 21 members; Thanked: 158 times

by pemdas » Sun Jul 10, 2011 12:43 am

the sum of the interior angles divided by the number of sides
180(n-2)/n where n is the number of sides in the polygon --> 180(8-2)/8=135;

y=180-135=45; AB=BC hence z=(180-135)/2=45/2
y-z=45-45/2=22.5`
x=180-2*(45/2)=135

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MBA.Aspirant: Master | Next Rank: 500 Posts; Posts: 401; Joined: Tue May 24, 2011 1:14 amFollowed by: 5 members; Thanked: 37 times

by MBA.Aspirant » Sun Jul 10, 2011 12:51 am

artstudent wrote:

The answer is 135 degrees

we know an octagon have equal sides and each of its interior angles = 135

if you look at one of the triangles formed you'll see that 2 of its sides are formed of the octagon sides i.e it's isosceles triangle

each of its smaller angles then = 180-135 /2 = 22.5

when the two triangles intersect they form a smaller triangle (the one opposite the x angle). each of the small angles in this triangle are common with the other 2 triangles = 22.5

so the angle opposite X = 180 - 22.5 - 22.5 = 135

and since they're both equal it follows that x = 135

Last edited by MBA.Aspirant on Sun Jul 10, 2011 12:53 am, edited 1 time in total.

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artstudent: Senior | Next Rank: 100 Posts; Posts: 62; Joined: Mon Mar 28, 2011 9:35 pm; Thanked: 4 times

by artstudent » Sun Jul 10, 2011 12:52 am

can you elaborate on the 180(8-2)/8=135, what angle is 135? exterior? I don't get how your getting y=180-135

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MBA.Aspirant: Master | Next Rank: 500 Posts; Posts: 401; Joined: Tue May 24, 2011 1:14 amFollowed by: 5 members; Thanked: 37 times

by MBA.Aspirant » Sun Jul 10, 2011 12:56 am

To get the interior angle of any polygon is = n-2 * 180 /n, where n is the no. of its sides

so in this case, 8-2 * 180 /8 = 135

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artstudent: Senior | Next Rank: 100 Posts; Posts: 62; Joined: Mon Mar 28, 2011 9:35 pm; Thanked: 4 times

by artstudent » Sun Jul 10, 2011 12:56 am

Perphaps I don't understand the definition of octagon but are all the sides equal in an octagon? I thought octagon just means 8 sides. Length is whatever.

MBA.Aspirant wrote:
artstudent wrote:
The answer is 135 degrees

we know an octagon have equal sides and each of its interior angles = 135

if you look at one of the triangles formed you'll see that 2 of its sides are formed of the octagon sides i.e it's isosceles triangle

each of its smaller angles then = 180-135 /2 = 22.5

when the two triangles intersect they form a smaller triangle (the one opposite the x angle). each of the small angles in this triangle are common with the other 2 triangles = 22.5

so the angle opposite X = 180 - 22.5 - 22.5 = 135

and since they're both equal it follows that x = 135

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MBA.Aspirant: Master | Next Rank: 500 Posts; Posts: 401; Joined: Tue May 24, 2011 1:14 amFollowed by: 5 members; Thanked: 37 times

by MBA.Aspirant » Sun Jul 10, 2011 1:06 am

yes for a regular octagon, like this one, all sides are equal and all interior angles are equal.

Check this: https://www.mathsisfun.com/geometry/regu ... ygons.html

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artstudent: Senior | Next Rank: 100 Posts; Posts: 62; Joined: Mon Mar 28, 2011 9:35 pm; Thanked: 4 times

by artstudent » Sun Jul 10, 2011 1:07 am

got it thanks.

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