10000 diagonals

Geometry Level 2

let f(x) = the number of diagonals in a x-gon polygon

find f(10000)


The answer is 49985000.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Nicolas Bryenton
Jul 23, 2014

Let us consider one individual vertex. A diagonal is a line formed when that vertex and another point are joined. However, if a point is directly adjacent to it, the line formed will not be a diagonal (it will be an edge). Also, it is obvious that the point itself cannot be chosen (no line formed). Therefore a vertex will be formed between that vertex, and all but three (one on each side and the point itself) of the other vertices. This is true for all the vertices. Thus,

f ( x ) = x ( x 3 ) f(x)=x(x-3) (so far)

However, connecting vertices A and B forms the same diagonal as connecting B and A. We are therefore counting each individual diagonals as two distinct diagonals, which is not true, so we must divide by 2. Therefore our function becomes

f ( x ) = 1 / 2 x ( x 3 ) f(x)=1/2*x(x-3)

and finally,

f ( 10 , 000 ) = 49985000 f(10,000)=49985000

Rakshit Pandey
Jul 31, 2014

Number of diagonals for a polygon with x x sides is given by:
f ( x ) = x ( x 3 ) 2 f(x)= \frac{x(x-3)}{2}
In this case, x = 10000 x=10000 .
So,
f ( x ) = x ( x 3 ) 2 f(x)= \frac{x(x-3)}{2}
f ( 10000 ) = 10000 × ( 10000 3 ) 2 \Rightarrow f(10000)= \frac{10000\times (10000-3)}{2}
f ( 10000 ) = 10000 × 9997 2 \Rightarrow f(10000)=\frac{10000\times 9997}{2}
f ( 10000 ) = 99970000 2 \Rightarrow f(10000)=\frac{99970000}{2}
f ( 10000 ) = 49985000 \Rightarrow f(10000)=49985000
So, number of diagonals is 49985000 \boxed {49985000} .



Math Man
Apr 3, 2014

actually f(x) = x(x-3)/2

then its vry easy to vertify that f(10000) = 10000 x 9997 / 2 = 49985000

typo vry very

math dude - 7 years, 2 months ago

Did the same!

Anik Mandal - 7 years, 1 month ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...