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If DE is parallel to AC, and point D is halfway between points A and B

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Joined: 02 Sep 2009
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If DE is parallel to AC, and point D is halfway between points A and B [#permalink]

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06 Nov 2017, 21:22
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45% (medium)

Question Stats:

74% (00:49) correct 26% (00:47) wrong based on 34 sessions

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If DE is parallel to AC, and point D is halfway between points A and B, what is the ratio of the area of triangle DBE to the area of triangle ABC?

(A) 2 :3
(B) 1 : 2
(C) 1 : 3
(D) 1 : 4
(E) 1 : 5

[Reveal] Spoiler:
Attachment:

2017-11-07_0920.png [ 20.49 KiB | Viewed 441 times ]
[Reveal] Spoiler: OA

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Kudos [?]: 135364 [0], given: 12691

Director
Joined: 25 Feb 2013
Posts: 619

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Location: India
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Re: If DE is parallel to AC, and point D is halfway between points A and B [#permalink]

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08 Nov 2017, 06:51
1
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Bunuel wrote:

If DE is parallel to AC, and point D is halfway between points A and B, what is the ratio of the area of triangle DBE to the area of triangle ABC?

(A) 2 :3
(B) 1 : 2
(C) 1 : 3
(D) 1 : 4
(E) 1 : 5

[Reveal] Spoiler:
Attachment:
2017-11-07_0920.png

As DE is parallel to AC and passes through the mid point of AB so as per Mid-point theorem $$DE=\frac{1}{2}AC$$

and triangle ABC is similar to triangle DBE

if the height of triangle ABC is $$h$$ then height of triangle DBE $$= \frac{h}{2}$$

So area of triangle DBE $$= \frac{1}{2}*\frac{h}{2}*\frac{AC}{2}$$

and area of triangle ABC $$= \frac{1}{2}*h*AC$$

so the ratio of areas $$= 1:4$$

Option D

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Re: If DE is parallel to AC, and point D is halfway between points A and B [#permalink]

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08 Nov 2017, 11:06
If we assume ∆abc as equilateral with base x and height x√3 area will be x^2 √3 / 2

Area of BDE will be x^2√3 / 8

So are they in the ratio of 1:3

Am i right or wrong?

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Director
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If DE is parallel to AC, and point D is halfway between points A and B [#permalink]

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08 Nov 2017, 11:23
jrk23 wrote:
If we assume ∆abc as equilateral with base x and height x√3 area will be x^2 √3 / 2

Area of BDE will be x^2√3 / 8

So are they in the ratio of 1:3

Am i right or wrong?

Sent from my Redmi Note 3 using GMAT Club Forum mobile app

Hi jrk23

the highlighted part is incorrect. if side of the equilateral side is x, then the sides of smaller triangle will be halved i.e x/2
so area of smaller circle will be $$x^2\sqrt{3}/4*4$$
and area of bigger circle will be $$x^2\sqrt{3}/4$$

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Joined: 17 May 2016
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Location: India
Schools: ISB '19 (I), IIMA (I)
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Re: If DE is parallel to AC, and point D is halfway between points A and B [#permalink]

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08 Nov 2017, 11:37
1
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Ratio of areas of similar triangles is equal to ratio of squares of corresponding sides.

∆ABC~∆DBE Since angle B is common and DE||AC, So angle BAC = Angle BDE

AAA similarity.

AB = 2DB

Ratio of areas = DB^2/AB^2
= 1/4
= 1:4

Option D

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Re: If DE is parallel to AC, and point D is halfway between points A and B   [#permalink] 08 Nov 2017, 11:37
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