Geometry Problem. Post your solution in the comment box below.

Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to view more details of problem 1279.

## Friday, October 28, 2016

### Geometry Problem 1279 Parallelogram, Midpoint, Triangle, Area, 3/10

Labels:
area,
midpoint,
parallelogram,
triangle

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https://goo.gl/photos/L9wsdrtPF7pDebWf7

ReplyDeleteLet DE extended meet BC at P

Since EB= ½. CD => PB=BC=AD

Triangle GAD similar to GFP => GA/GF=AD/PF= 2/3=> GA/AF=2/5 => S(DGF)/S(ADF)=3/5

S(DGF)=3/5.S(ADF)= 3/5 x ½ S(ABCD)= 3/10. S(ABCD)

Let M be the mid point of AF.

ReplyDeleteEM = AD/4 so AD = 4/5AM

Now S(AEM) = 1/4 S(ABF) = 1/4(S/4) = S/16

Hence S(AED) = 4/5 X S/16 = S/20

S1 = S -S(ABF) - S(ADE) - S(CDF) + S(ADE)

Hence S1 = S -S/4 - S/4 - S/4 + S/20 = 3S/10

Sumith Peiris

Moratuwa

Sri Lanka

In the 2nd line it should read as

DeleteAG = 4/5AM not AD = 4/5AM

S(AED) = S/20 ??

ReplyDeleteS(AEM) = 1/4 S(ABF) = 1/4(S/4) = S/16

DeleteHence S(AED) = 4/5 X S/16 = S/20

Sumith

DeletePer observation, S(AED) is not equal S/20.

Peter Tran

Area of triangle AED is 1/4 the area of ABCD, because triangle has same base as ABCD and height 1/2 of height of parallelogram ABCD

DeleteExtremely sorry Omid and Peter. My mistake.

DeleteI should have said S(AEG) = S/20 which is easily seen since S(AEM) = S/16 and AG = 4/5AM

Extend AF to meet DC at P

ReplyDeleteSince FC is half of AD, from mid-point theorem we have

CP = CD & FP = AF = l (say) and AG = x (say)

Also the triangles DGP and EGA are similar

=> DP/EA = PG/AG

=> 4 = 2l-x/x

=> x = AG = (2/5)l and GF = (3/5)l

Since the area of triangle AFD = 1/2(S) => Area of triangle GFD (S1) = 3/5*1/2*S (S being the are of //gm ABCD)

Therefore S1 = 3/10*S

Draw CG, G midpoint of AD, H on ED. Tr GHD ~ tr AEG => S(AEG)=1/20 (S)

ReplyDeleteAGD + AEG + EGFB + FCD + S1 = S => 1/5 + 1/20 + 1/5 + 1/4 + S1 = S

S1 = 7/10 S

Another synthetic proof : See graph here

ReplyDeleteLet FH // AB intercept ED in I, H on AD

IH = AE/2 = AB/4 => FI = 3/4.AB = 3/2.AE

ΔAEG and ΔFIG are similar, ratio 2/3

AF = AG + GF = 5/2.AG => AG = 2/5.AF

S(AGE) = 2/5.S(AFE) = 2/5.1/2.S(AFB) = 1/5.1/4.S = 1/20.S

S(AGD) = S(AED) – S(AGE) = [1/4 - 1/20].S = 1/5.S

S1 = S(AFD) – S(AGD) = [1/2 - 1/5].S = 3/10.S QED

I posted here another nice solution, by José Luis da Vila, that I found on Twitter - with a little bit more details than the original.

ReplyDelete