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# In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a

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Manager
Joined: 19 Dec 2016
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In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a [#permalink]

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02 Aug 2017, 19:00
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In the diagram DE is parallel to AC. If AC = 10 and DE = 5 and the area of triangle ABC is 40, then what is the area of triangle BDE?.

(A) 8
(B) 10
(C) 12
(D) 14
(E) 20
[Reveal] Spoiler: OA

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Last edited by GTExl on 03 Aug 2017, 10:50, edited 2 times in total.
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Re: In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a [#permalink]

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02 Aug 2017, 19:43
GTExl, you have missed the answer options!
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Re: In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a [#permalink]

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02 Aug 2017, 19:58
pushpitkc wrote:
GTExl, you have missed the answer options!

Thanks pushpitkc. i have corrected that.
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In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a [#permalink]

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03 Aug 2017, 12:28
GTExl wrote:
Attachment:
Capture.PNG

In the diagram DE is parallel to AC. If AC = 10 and DE = 5 and the area of triangle ABC is 40, then what is the area of triangle BDE?.

(A) 8
(B) 10
(C) 12
(D) 14
(E) 20

The two triangles are similar, so the ratio of one side to another will hold for all lengths.

The triangles are similar because they have three identical angles. They share angle B.

Segments AC and DE are parallel (given), cut by two traversal segments AB and BC.

Therefore corresponding angles C and E are congruent, as are corresponding angles A and D. AAA = similar triangles.

Base of large triangle = 10, base of smaller triangle = 5. Ratio of lengths (not area) is 2:1.

We need height of large triangle to calculate height and area of smaller triangle. Obtain larger triangle's height from larger triangle's given area.

Area = $$\frac{1}{2}b * h$$

40 = $$\frac{1}{2}$$*(10)*(h)
h = 8

Large:small is 2:1 for any one-dimensional length. Height of large triangle is 8; height of small triangle is therefore 4.

Area of smaller triangle BDE is
$$\frac{1}{2}$$*5*4 = 10

Alternatively, take scale factor 2, call it K. For areas, use K$$^2$$ (because area = length*length, scale factor is deployed twice, K*K).

Area of 40 = 2$$^2$$ * area of smaller triangle

40 = 4 * smaller triangle's area

10 = area of smaller triangle BDE

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Re: In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a [#permalink]

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06 Aug 2017, 05:16
Area of traingle DBE = AREA OF triangle ABC - AREA of trapezoid DEAC

let a be the height of triangle BDE and b be the height of trapezoid then height of triangle will be (a+b)

1/2 x 5 x a = 40 - 1/2 (10 + 5) (8-a)

a=4

area = 10
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Re: In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a [#permalink]

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06 Aug 2017, 06:39
1
KUDOS
ratio of the area of similar triangles = ration of the square of corresponding sides

so x = (25/100 )* 40

= 10
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In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a [#permalink]

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09 Aug 2017, 12:15
I think this can be done fairly quickly using unit of measure principles.

When talking about sides only the scale up and down is linear.

So for similar triangles, if the side is 10 for the larger triangle and 5 for the smaller triangle, the scale down is by 1/2 when comparing the sides.

For area the scale down would be the square of the scale down of the side = (1/2)^2 = 1/4 of the larger area
1/4 * 40 = 10.

The scale up/down factor is to the power of 1 for sides, power of 2 for area and power of 3 for volume.

Ans is B.
In the following diagram DE is parallel to AC. If AC = 10 and DE = 5 a   [#permalink] 09 Aug 2017, 12:15
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