Proposed Problem

Click the figure below to see the complete problem 410 about Two Regular Pentagons, Angle.

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Complete Problem 410

Level: High School, SAT Prep, College geometry

## Sunday, December 27, 2009

### Problem 410: Two Regular Pentagons, Angle

Labels:
angle,
diagonal,
pentagon,
regular polygon

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http://geometri-problemleri.blogspot.com/2009/12/problem-58-ve-cozumu.html

ReplyDeletethe two pentagons are similar ,by the similarity s,with center D,angle (108+m(CDF)),ratio GH/AB

ReplyDeletes(B)=H;s(A)=G

x=108+m(CDF)

.-.

To Anonymous:

ReplyDeleteThe answer is wrong.

in the complex plane we can choose O center from the circumcircle of ABCDE as origin,affix of D

ReplyDeletezD=1,then:

zC= exp(i2pi/5);zB= exp(i4pi/5);zA= exp(i6pi/5)

the similarity s can be written: z'=az+b

s(D)=D b=1-a

zG=azA+1-a;zH=azB+1-a

ang(HMG)=arg((zG-zA)/(zH-zB))

=arg[(exp(i4pi/5)-1)/(exp(i6pi/5)-1)]

=arg[exp(ipi/5).sin(2pi/5)/sin(3pi/5)]=pi/5

in degrees

x=180-36=144

.-.

The problem is simple if you assume that there is an answer to the problem. It is posed in a way that suggests that the answer is independent of the angle between the sides of the pentagon. If I assume that there is a unique answer to the problem, then I can orient the pentagons any way I choose without loss of generality. So I choose to orient them so that angle x occurs at D. Then x is simply the exterior angle of the well-know 36-72-72 triangle, and thus must measure 144 degrees.

ReplyDeletehttp://img577.imageshack.us/img577/1528/problem410.png

ReplyDeleteNote that triangles ADG and BDH are congruence ( case SAS)

∠ (XMD)= ∠ (MHD) => quadrilateral MGHD is cyclic

And ∠ ( GMX)= ∠ (GDH) = 36 ----- ( both angles face same arc GH)

And x= 180-36=180

Problem 410

ReplyDeleteThe circle described in both pentagons intersect at point Ν , then <ANB=36=<ANE=

=<END=<DNJ=<JNH ,but <BNG=36+36+36+36+36=180.Therefore section B, N, C and A, N, H is collinear so points M, N coincide.So <BMG=144.(<GMH=36).

APOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORYDALLOS PIRAEUS GREECE