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Math Expert V
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If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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16 00:00

Difficulty:   35% (medium)

Question Stats: 75% (02:21) correct 25% (02:26) wrong based on 487 sessions

### HideShow timer Statistics Tough and Tricky questions: Geometry.

Attachment: 2014-10-28_1955.png [ 26.86 KiB | Viewed 9995 times ]
If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, what fraction of the trapezoid BACE is shaded?

A. 1/5
B. 1/4
C. 1/3
D. 3/8
E. 1/2

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Re: If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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Let Height of Triangle = Height of Trapezoid = h

Area of triangle $$= \frac{1}{2} * 6 * h$$

Area of trapezoid $$= \frac{1}{2} (6+18)*h$$

Ratio $$= \frac{6}{24} = \frac{1}{4}$$

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If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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Triangle ABD area equal: 6* square root 3/2 * 6/2 = 9*square root of 3
ABCE area equal: Height * ( AB+ CE)/2 = Height* ( 6+18)/2 = Height * 12.
Height = 6* square root of 3/2
ABCE Area = 36*squareroot of 3
=> ABD area/ABCE area =1/4

There maybe a short cut approach?
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If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Geometry.

Attachment:
The attachment 2014-10-28_1955.png is no longer available
If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, what fraction of the trapezoid BACE is shaded?

A. 1/5
B. 1/4
C. 1/3
D. 3/8
E. 1/2

This is a isosceles trapezium, however I'm facing difficulty to prove it via calculation

Attachment: 2014-10-28_1955.png [ 55.26 KiB | Viewed 9631 times ]

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Re: If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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Area of a Trapezium = 1/2 * ( Sum of lengths of Two parallel Lines) * Distance between two Lines.

Distance between two lines = height of equilateral Triangle.

So Answer = area of Equilateral Triangle / Area of Trapezium = 1/4.

Hope This Helps!!!! Regards
Narayana Raju G
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Posts: 2821
Re: If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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Quote:

Attachment:
2014-10-28_1955.png
If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, what fraction of the trapezoid BACE is shaded?

A. 1/5
B. 1/4
C. 1/3
D. 3/8
E. 1/2

We can let the height of the triangle (which is also the height of the trapezoid) = h.

Let’s first determine the area of triangle ABD in terms of h:

area = (1/2)(base)(height) = (1/2)(6)(h) = 3h

Next we can determine the area of trapezoid BACE in terms of h:

area = (1/2)(base 1 + base 2)(height) = (1/2)(18 + 6)(h) = 12h

Thus, the fraction of the trapezoid that is shaded is 3h/12h = 1/4.

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Re: If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Geometry.

Attachment:
The attachment 2014-10-28_1955.png is no longer available
If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, what fraction of the trapezoid BACE is shaded?

A. 1/5
B. 1/4
C. 1/3
D. 3/8
E. 1/2

Although Area of Trapezoid formula will make this easier, it is not required. I thought I'd post the way to do this without that formula:

Method Using Only Triangle Formulas

AB=6, and that triangle is equilateral, so AB=BD=DA=6.

Our height will be 3$$\sqrt{3}$$. We can make Right Triangles and determine this. (See picture)

Height = 3$$\sqrt{3}$$ for all three triangles

Area of ABD = $$\frac{1}{2}$$ * b * h ----> $$\frac{1}{2}$$ * 6 * 3$$\sqrt{3}$$ ----> 9$$\sqrt{3}$$

OTHER TWO TRIANGLES

We know that CAD=DBE. They are similar triangles. It is given that CE=18, so CD=DE=9.

Base = 9

Area of CAD = $$\frac{1}{2}$$ * 9 * h

Area of CAD = $$\frac{1}{2}$$ * 9 * 3$$\sqrt{3}$$ = $$(27\sqrt{3})/2$$

We have two of these triangles, so we multiply $$(27\sqrt{3})/2$$ by 2 and get 27$$\sqrt{3}$$

TOTAL AREA OF ALL THREE TRIANGLES

27$$\sqrt{3}$$ + 9$$\sqrt{3}$$ = 36$$\sqrt{3}$$

9$$\sqrt{3}$$ / 36$$\sqrt{3}$$ = $$\frac{1}{4}$$

Attachments Trapezoid.jpg [ 95.06 KiB | Viewed 2230 times ]

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Re: If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha  [#permalink]

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Attachment: IMG_20180823_182625.jpg [ 58.75 KiB | Viewed 2192 times ]

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The best thing in life lies on the other side of the pain. Re: If triangle ABD is an equilateral triangle and AB = 6 and CE = 18, wha   [#permalink] 23 Aug 2018, 05:55
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