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

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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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New post 06 Nov 2017, 21:22
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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
2017-11-07_0920.png [ 20.49 KiB | Viewed 441 times ]
[Reveal] Spoiler: OA

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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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New post 08 Nov 2017, 06:51
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Bunuel wrote:
Image
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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New post 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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If DE is parallel to AC, and point D is halfway between points A and B [#permalink]

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New post 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?

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

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New post 08 Nov 2017, 11:37
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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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If DE is parallel to AC, and point D is halfway between points A and B

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