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In the rectangle above, A is the midpoint of the side, and

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DeeptiM
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In the rectangle above, A is the midpoint of the side, and [#permalink] New post  Updated on: 12 Jun 2019, 02:18
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In the rectangle above, A is the midpoint of the side, and BC = CD = DE. What is the area of the rectangle?

(1) The area of the shaded region is 24.
(2) The area of triangle CDO is 16.

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Originally posted by DeeptiM on 15 Aug 2011, 01:57.
Last edited by Bunuel on 12 Jun 2019, 02:18, edited 2 times in total.
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In the rectangle above, A is the midpoint of the side, and [#permalink] New post  31 Jul 2014, 06:33
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In the rectangle above, A is the midpoint of the side, and BC = CD = DE. What is the area of the rectangle?

The area of a rectangle is (width)*(length).

Notice that since A is the midpoint of the width of the rectangle, then AB = (width)/2
Also, since BC = CD = DE, then CD = (length)/3.

(1) The area of the shaded region is 24. The area of the shaded triangle = 1/2*BE*AB = 1/2*(length)*(width)/2 = 24 --> (length)*(width) = 96. Sufficient.

(2) The area of triangle CDO is 16. The area of triangle CDO = 1/2*OE*CD = 1/2*(width)*(length)/3 = 16 --> (width)*(length) = 96. Sufficient.

Answer: D.

Hope it's clear.
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eragotte
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Re: In the rectangle above, A is the midpoint of the side, and [#permalink] New post  15 Aug 2011, 05:58
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Consider this, area for triangle is BH/2.

the two bigger triangles have the same B & H, the three small triangles have the same base and height. thus we can say the three smaller triangles or the 2 larger are equal. with (1) or (2) we can see the area of the full rectangle is 96.

(18*3)*2 or (24*2)*2
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Re: In the rectangle above, A is the midpoint of the side, and [#permalink] New post  15 Aug 2011, 06:10
well u wld hve to b a lil patient wid me...

somehow um still unclear with the explanation :(
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Re: In the rectangle above, A is the midpoint of the side, and [#permalink] New post  15 Aug 2011, 07:08
DeeptiM wrote:
well u wld hve to b a lil patient wid me...

somehow um still unclear with the explanation :(


D from my side.

1. Area of rectangle = 2*2*Area(AOB) - Suff.
2. Area of rectangle = 2*3*Area COD (because Area BOC= Area COD= Area DOC) - Suff.

Hope it helps.

Cheers,
Aj.
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Re: In the rectangle above, A is the midpoint of the side, and [#permalink] New post  15 Aug 2011, 07:40
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DeeptiM wrote:
In the rectangle above, A is the midpoint of the side, and BC=CD=DE. What is the area of the rectangle?
(1) The area of the shaded region is 24.
(2) The area of triangle CDO is 16.

I got the answer but need a more concise approach..

Attachment:
Geomtery_Median_DS.JPG


Let's call the vertex diagonally opposite of E as F.

1.
OA is the median of \(\triangle OBF\), because A is the mid-point of FB. A median divides a triangle in two-halves such that the areas of two newly formed triangles are equal.

Area(OAF)=Area(OAB)=24
Area(OBF)=2*Area(OAB)=2*24=48

A diagonal of a rectangle divides the rectangle in two equal halves such that the area remains same for both halves.

Thus, Area(ABEO)=2*Area(BOF)=2*48=96
Sufficient.

2.
Same concept as statement 1.
OD is the median of OCE because CD=DE
AND
OC is the median of OBD because CD=BC

Area(ODE)=Area(OCD)=Area(OCB)=16
Area(OBE)=3*16=48
Area(FBEO)=2*Area(OBE)=2*48=96
Sufficient.

Ans: "D"
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Re: In the rectangle above, A is the midpoint of the side, and [#permalink] New post  12 Jun 2019, 02:15
Vijayeta wrote:
Need a explanation of this.

https://pasteboard.co/Ij3uxKD.jpg (click the url to see my solution)

Just labelling the sides of the rectangle as 2x (for the side having point A) and 3y (for the side having pts B,C,D) does the trick. No complex calculation needed in this question.
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Re: In the rectangle above, A is the midpoint of the side, and [#permalink] New post  16 Mar 2023, 23:42
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