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

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Math Expert
Joined: 02 Sep 2009
Posts: 49298
If DE is parallel to AC, and point D is halfway between points A and B  [#permalink]

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06 Nov 2017, 22:22
00:00

Difficulty:

25% (medium)

Question Stats:

76% (00:56) correct 24% (01:18) wrong based on 50 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

Attachment:

2017-11-07_0920.png [ 20.49 KiB | Viewed 760 times ]

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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, 07:51
1
1
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

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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Joined: 26 Sep 2017
Posts: 28
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, 12:06
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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08 Nov 2017, 12:23
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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Joined: 18 May 2016
Posts: 182
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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, 12:37
1
1
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 &nbs [#permalink] 08 Nov 2017, 12:37
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