Awesome geometry - 3

Geometry Level 4

In parallelogram A B C D ABCD shown above, E E is the midpoint of B C BC , F F is the midpoint of C D CD . G G is a point on E F EF such that ratio E G : G F EG : GF equal 1 : 2 1 : 2 . If the area of this parallelogram is 776, find Area of triangle A E G AEG .


The answer is 97.

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Nihar Mahajan by Nihar Mahajan , 20, India

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3 solutions

A r e a A B E = A F D = 2 F E C = 1 / 4 ∗ A B C D . A r e a A E F = A r e a s A B C D − ( A B E + A F D + F E C = 3 / 8 ∗ A B C D A r e a A E G = 1 / 3 ∗ A E F = 1 / 8 ∗ A B C D = 776 / 8 = 97. Area~ABE=AFD=2FEC=1/4*ABCD.\\ Area~AEF=Areas~ABCD-(ABE+AFD+FEC=3/8*ABCD\\ Area~AEG=1/3*AEF=1/8*ABCD=776/8=97. \\

D e t a i l s : − A r e a ∣ ∣ g r a m A B C D = B C ∗ H e i g h t . A r e a Δ A B E = 1 2 ∗ B C 2 ∗ H e i g h t = 776 / 4. A r e a Δ A F D = 1 2 ∗ H e i g h t 2 ∗ A D = 1 2 ∗ H e i g h t 2 ∗ B C = 776 / 4. A r e a Δ D E C = 1 2 ∗ B C 2 ∗ H e i g h t = 776 / 4. B u t A r e a Δ F E C = 1 2 ∗ A r e a Δ D E C = 776 / 8. B a s e i s 1 / 2. S o A r e a Δ A E F = ( 1 − 1 / 4 − 1 / 4 − 1 / 8 ) ∗ 776. B u t A r e a Δ A E G = 1 3 Δ A E F = 1 3 ∗ 3 8 ∗ 776 = 97. B a s e i s 1 / 3. Details:-\\ Area~| |gram~ABCD=BC*Height.\\ Area~\Delta~ABE=\frac 1 2*\dfrac{BC} 2*Height=776/4.\\ Area~\Delta~AFD=\frac 1 2*\dfrac{Height} 2*AD=\frac 1 2*\dfrac{Height} 2*BC=776/4.\\ Area~\Delta~DEC=\frac 1 2*\dfrac{BC} 2*Height=776/4.\\ But~Area~\Delta~FEC=\frac 1 2*Area~\Delta~DEC=776/8. ~~~Base~ is~ 1/2.\\ So~Area~\Delta~AEF=(1-1/4-1/4-1/8)*776.\\ But~Area~\Delta~AEG~=\frac 1 3~\Delta~AEF=\frac 1 3*\frac 3 8*776=97.~~~Base~is~1/3.

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Kenny Lau
Aug 15, 2015

We shall calculate the area of AEF first. (Refer to image 1)

Note that [AEF] = [AHE] + [AHF] + [HEF].

Then, [AHE] can be slided to [JHE] and [AHF] to [KHF]. (Refer to image 2)

Now the area is clearly 3 8 \frac38 of the whole parallelogram.

Therefore, [AEF] = 3 8 × \frac38\times [ABCD] = 291.

Then, AG divides AEF into two areas of 1:2.

Therefore, [AEG] = 1 1 + 2 × \frac1{1+2}\times [AEF] = 97.

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Praful Jain
May 29, 2015

AREA OF BCD IS 1/2 ABCD AREA OF EFC IS 1/4 ABCD [MID POINT THEROM] AREA OF EGC IS 1/8 ABCD[MID POINT THEROM] AREA OF AEC IS 1/4 ABCD[MID POINT THEROM] AREA OF EGC-AREA OF AEC=AREA OF AEG 1/4-1/8=1/8 1/8 OF 776 IS EQUAL TO 97 IS THE AREA OF AEG

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