Proposed Problem

Click the figure below to see the complete problem 398 about Parallelogram, Area, Similarity, Congruence.

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Complete Problem 398

Level: High School, SAT Prep, College geometry

## Saturday, December 5, 2009

### Problem 398: Parallelogram, Area, Similarity, Congruence

Labels:
area,
congruence,
parallelogram,
similarity,
triangle

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ReplyDeleteMN mark as x. Express EM and NC by x (thales theorem)

Express h1 (altitude EBC, or AFD) by h2 (altitude AECF)

(similar triangles)

Express S=s1+s2+s3 (s1 EBC, s2 AECF, s3 AFD)

Let S2= Area(BHDG)

ReplyDeleteWe have NG=HQ=2/3 *PD [Tri. AQH~Tri. APD]

NP=1/2* PD [Tri. DPF~ Tri. DNC]

So NG/NP=4/3

Calculate DG/NP= (PD+NP+NG)/NP= 2+1+4/3=13/3

S1/S2=Area(MNPQ)/Area(BHDG)=NP/GD =3/13 [Both para. have the same height]

S2/S= Area(BHDG)/Area(ABCD)= DH/DA=1/3 [Both para. have the same height]

So S1/S=1/13

Peter Tran

vstran@yahoo.com