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Senior Manager  P
Joined: 19 Oct 2018
Posts: 465
Location: India
BCDF is a rectangle. Triangle ABE has an area of 2 cm^2. Triangle BEF  [#permalink]

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4 00:00

Difficulty:   75% (hard)

Question Stats: 18% (03:33) correct 82% (02:53) wrong based on 11 sessions

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BCDF is a rectangle. Triangle ABE has an area of 2$$cm^2$$ . Triangle BEF has an area of $$3cm^2$$. Find the area of the blue region.

A. 4
B. 4.5
C. 5
D. 5.5
E. 6

Attachments image 1.png [ 17.41 KiB | Viewed 363 times ]

Senior Manager  P
Joined: 19 Oct 2018
Posts: 465
Location: India
Re: BCDF is a rectangle. Triangle ABE has an area of 2 cm^2. Triangle BEF  [#permalink]

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1
Area of Blue portion + Area of green portion = Area of Grey portion + Area of yellow portion = Half the area of rectangle
We know the area of green portion and yellow portion. Now, if we able to find the area of Grey portion, we can easily find the area of Blue portion.

Triangle ABE is similar to the triangle to FDE
Angle BAE= Angle DFE and angle ABE = angle EDF {alternate interior angles}
Hence, $$\frac{Area ABE}{Area of FDE}$$ = $$\frac{AE^2}{EF^2}$$......(1)

Also, Triangle ABE and Triangle BEF has same base AE and EF respectively
Hence $$\frac{Area ABE}{Area BEF}$$= $$\frac{AE}{EF}$$
$$\frac{2}{3}$$=$$\frac{AE}{EF}$$
Put value of AE/EF in equation (1)
We get, $$\frac{Area ABE}{Area of FDE}$$= $$\frac{4}{9}$$
$$\frac{2}{Area of FDE}$$ = $$\frac{4}{9}$$
Area of FDE= 4.5 { area of grey portion}

Area of Blue portion + Area of green portion = Area of Grey portion + Area of yellow portion
Area of Blue portion= 4.5+3-2=5.5
nick1816 wrote:
BCDF is a rectangle. Triangle ABE has an area of 2$$cm^2$$ . Triangle BEF has an area of $$3cm^2$$. Find the area of the blue region.

A. 4
B. 4.5
C. 5
D. 5.5
E. 6
Director  V
Joined: 27 May 2012
Posts: 800
BCDF is a rectangle. Triangle ABE has an area of 2 cm^2. Triangle BEF  [#permalink]

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nick1816 wrote:
BCDF is a rectangle. Triangle ABE has an area of 2$$cm^2$$ . Triangle BEF has an area of $$3cm^2$$. Find the area of the blue region.

A. 4
B. 4.5
C. 5
D. 5.5
E. 6

Hi nick1816,
Thank you for your question and Soln. but the below part is not clear to me, can you help? Which same base are you talking about?Thanks.

nick1816 wrote:

Also, Triangle ABE and Triangle BEF has same base AE and EF respectively
Hence $$\frac{Area ABE}{Area BEF}$$= $$\frac{AE}{EF}$$
$$\frac{2}{3}$$=$$\frac{AE}{EF}$$

Attachment: image 1.png [ 8.72 KiB | Viewed 113 times ]

_________________
- Stne
Senior Manager  P
Joined: 19 Oct 2018
Posts: 465
Location: India
BCDF is a rectangle. Triangle ABE has an area of 2 cm^2. Triangle BEF  [#permalink]

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Draw a perpendicular from BN on line AF

Area of BAE= 1/2 * AE * BN
Area of BEF= 1/2 * EF * BN

Area of BAE/ Area of BEF= (1/2 * AE * BN) / (1/2 * EF * BN)= AE/EF
Attachments 2.png [ 7.31 KiB | Viewed 97 times ]

Director  V
Joined: 27 May 2012
Posts: 800
Re: BCDF is a rectangle. Triangle ABE has an area of 2 cm^2. Triangle BEF  [#permalink]

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nick1816 wrote:
Draw a perpendicular from BN on line AF

Area of BAE= 1/2 * AE * BN
Area of BEF= 1/2 * EF * BN

Area of BAE/ Area of BEF= (1/2 * AE * BN) / (1/2 * EF * BN)= AE/EF

So I guess in your first post you meant BAE and BEF have same height,and their bases lie on the same straight line. Got it.
Also in this post , I am sure you mean , draw a perpendicular BN on AF Got it.

Thanks a ton. +1
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- Stne Re: BCDF is a rectangle. Triangle ABE has an area of 2 cm^2. Triangle BEF   [#permalink] 23 May 2019, 11:18
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BCDF is a rectangle. Triangle ABE has an area of 2 cm^2. Triangle BEF  