http://www.blackdouglas.com.au/taskc...s/081pent1.gif

How can you show that a regular pentagon can be divided into 2010 triangles so that one is type A and 2009 are type B?

The middle triangle is type A.

The sides triangles are type B.

Thank you!

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- May 5th 2010, 11:20 PMGuMingPentagon problem
http://www.blackdouglas.com.au/taskc...s/081pent1.gif

**How can you show that a regular pentagon can be divided into 2010 triangles so that one is type A and 2009 are type B?**

The middle triangle is type A.

The sides triangles are type B.

Thank you! - May 6th 2010, 06:01 AMWilmer
Shouldn't that be "divided in 2011 (or 2009) triangles.....2010 (or 2008) are type B" ?

And what do you mean by "type B": similar triangles? - May 6th 2010, 10:27 AMSoroban
Hello, GuMing!

Quote:

Code:`A`

o

* : *

* *:* *

* ::: *

E o B *:::* B o B

* : A : *

* *:::::* *

* ::::::: *

o * * * o

D C

Show that a regular pentagon can be divided into 2010 triangles:

. . where 1 is type A and 2009 are type B.

Type A triangle has angles: .72°, 72°, 36°.

Type B triangle has angles: .36°, 36°, 108°.

In the above diagram, the pentagon is divided into: . 1 Type A, 2 Type B.

Consider the Type A triangle:

Code:`A`

*

/ \

/36°\

/ \

/ 2 \ F

/ 108° *

/ *72°\

/ * \

/36°* 1 \

/ * 36° 72° \

D o - - - - - - - - - o C

Bisect $\displaystyle \angle D.$

$\displaystyle \Delta DFC$ is a Type A triangle.

$\displaystyle \Delta AFD$ is a Type B triangle.

That is, a Type A triangle can be divided into: 1 Type A and 1 Type B.

We have divided the pentagon into: 1 Type A, 3 Type B.

Repeat the process with Type A triangle $\displaystyle DFC$

. . and we have: 1 Type A, 4 Type B.

and so on . . .

Therefore, starting with Type A triangle $\displaystyle ACD$,

. . perform the angle-bisection 2007 times

and we will have 1 Type A and 2009 Type B triangles.

- May 7th 2010, 04:19 AMGuMing
Thank you for your help(Clapping)(Talking)

- May 10th 2010, 08:49 AMChris L T521
The fact that (the years) "2010" and "2009" appear in the problem hint to us that this question may be from a maths competition.

This thread will remain closed as we conduct further investigation (we have**zero**tolerance for cheaters!!). If everything ends up being OK, it will be reopened.