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# If ABCD is a rectangle, what is the area of the shaded region?

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Math Expert
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If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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22 Jan 2015, 06:58
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Difficulty:

55% (hard)

Question Stats:

69% (02:48) correct 31% (02:47) wrong based on 367 sessions

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If ABCD is a rectangle, what is the area of the shaded region?

A. 64
B. 82
C. 96
D. 120
E. 150

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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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27 Jan 2015, 23:41
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Diagonal of rectangle $$= \sqrt{20^2 + 15^2} = \sqrt{625} = 25$$

Half the area of rectangle $$= 150 = \frac{1}{2} * 25 * h$$

Height = 12 = CP

Length BP $$= \sqrt{20^2 - 12^2} = \sqrt{256} = 16$$

Area of shaded region $$\triangle BPC = \frac{1}{2} * 16 * 12 = 96$$
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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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22 Jan 2015, 12:46
2
Area of Triangle BCD =1/2*20*15=150

BD= sqrt (20^2+15^2)=25
also area of BCD can be determined as 1/2* BD* height from C to BD
lets height from C meets BD at X
Area of triangle BCX will be
1/2* BD* CX =150
=>1/2*25* CX=150
=>CX=300/25=12

Now, BX= sqrt (20^2-12^2)=16
So, area of shaded region will be
1/2* BX*CX=1/2*16*12=96

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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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28 Jan 2015, 05:59
1
1
a way different from the one mentioned ....ans C
we can use similar triangles property between triangles abd and bcp and get the sides of triangle bcp and so area comes as 96..
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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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28 Jan 2015, 08:27
1
Here we go:

Consider triangle BCD

Area will be
Let the perpendiculer from C to BD be CM

1/2 * (CD*BC) = 1/2 * (CM * BD)

Solve

150 = 1/2 * (CM * 25) (BD is the diagonal---> Apply pythogorus Theorem)

CM = 12

Again apply pythagorus in shaded triangle

BM = 16

So Area will be 1/2 * (BM * CM) = 96

Option C is correct
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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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28 Jan 2015, 12:29
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2
Hi All,

When the GMAT puts a right triangle in front of you, it is probably NOT a random right triangle. While there are some exceptions, in most cases, you're looking at a 3/4/5, 5/12/13, 30/60/90 or 45/45/90 right triangle. If you see ANY clues that point to a specific right triangle, then you should think about how you can prove that the triangle IS what you think it is.

Here, we have 3 right triangles and they all appear to be multiples of a 3/4/5 (notice how each has a diagonal that is a MULTIPLE of 5).

Assuming they're all 3/4/5s, the sides SHOULD be....

Big triangle: 15/20/25
Middle triangle: 12/16/20
Small triangle: 9/12/15

From the picture, there are a couple of other things that need to "line up" to prove that we're correct:

1) One side of the "small" and "middle" triangles MUST be the same.
2) The "other" sides of each of those triangles MUST add up to 25 (the diagonal of the big triangle)

The common "shared" side would be 12, which means the "other" sides (16 and 9) add up to 25. So we have the EXACT situation that we thought we did. With knowledge of ALL of the side lengths, we can now answer any question that's asked.

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** Manager Joined: 03 May 2013 Posts: 66 If ABCD is a rectangle [#permalink] ### Show Tags 07 Jun 2015, 20:14 If ABCD is a rectangle, what is the area of the shaded region? plz open link for image A. 64 B. 82 C. 96 D. 120 E. 150 Kudos for a correct solution. This has been discussed over here , if-abcd-is-a-rectangle-what-is-the-area-of-the-shaded-region-191988.html the topic is locked there hence opening new thread experts please tell what wrong with this let x be the point from c to digonal BD now required area = area of rectangle - area of trangle BAD - area of trangle cxd angle CDx and XCD are 45 digree , hence area of trangle CDE = 1/2 x 15/root2 x 15/root2 = approx 46 required area = 300 - 150 - 46 = 104 Intern Joined: 29 Mar 2015 Posts: 44 Location: United States Re: If ABCD is a rectangle [#permalink] ### Show Tags 07 Jun 2015, 23:15 The problem with your solution is that angle CXD and XCB are not 45 degrees - if they were, CX would bisect BD, but it does not (unless ABCD were a rhombus). The correct solution: We are looking for $$\frac{CX*BX}{2}$$ We know: BD is 25 (3-4-5 right triangle * 5), Area of BCD = 150, and therefore CX = 12 (because $$\frac{CX*BD}{2} = 150$$) So therefore: $$BX^2 + 144 = 400$$-> $$BX^2 = 256$$ -> $$BX = 16$$ -> Area is $$\frac{16*12}{2} = 96$$ Intern Joined: 04 May 2014 Posts: 29 Re: If ABCD is a rectangle, what is the area of the shaded region? [#permalink] ### Show Tags 08 Jun 2015, 12:41 If ABCD is a rectangle, what is the area of the shaded region? A. 64 B. 82 C. 96 D. 120 E. 150 Kudos for a correct solution. Pythagorean Theorem to solve for BD = 20^2 + 15^2 = 225 + 400 = 625 BD = sqrt (625) = 25 All these triangles inscribed are proportionate, since BD is a diagonal of this rectangle Since BC = 20 and BD = 25, BC = = 4/5*BD So BX = 4/5 * 20 = 16 Solve for CX = A^2 + B^2 = C^2 256 + B^2 = 400 B^2 = 400-256 B^2 = 144 Sqrt(144) = 12 Area of shaded region = 12*16 / 2 = 12 * 8 = 96 Hence answer C Manager Joined: 03 May 2013 Posts: 66 Re: If ABCD is a rectangle, what is the area of the shaded region? [#permalink] ### Show Tags 08 Jun 2015, 18:04 bluesquare wrote: The problem with your solution is that angle CXD and XCB are not 45 degrees - if they were, CX would bisect BD, but it does not (unless ABCD were a rhombus). The correct solution: We are looking for $$\frac{CX*BX}{2}$$ We know: BD is 25 (3-4-5 right triangle * 5), Area of BCD = 150, and therefore CX = 12 (because $$\frac{CX*BD}{2} = 150$$) So therefore: $$BX^2 + 144 = 400$$-> $$BX^2 = 256$$ -> $$BX = 16$$ -> Area is $$\frac{16*12}{2} = 96$$ Hi, Thanks for prompt response, I an't saying angle CXD and angle XCB are 45 digrees, what i am saying angle CXD is 90 and angle CDX are 45 digrees, since DB is diognal of restangle hence in trangle CXD , angle CXD is 90 , CDX is 45 then XCD must be 180-90-45=45, by that CX and DX must be 15/root2, which is not inline with correct explanation Experts please explain e-GMAT Representative Joined: 04 Jan 2015 Posts: 2577 Re: If ABCD is a rectangle, what is the area of the shaded region? [#permalink] ### Show Tags 10 Jun 2015, 23:45 1 vipulgoel wrote: bluesquare wrote: The problem with your solution is that angle CXD and XCB are not 45 degrees - if they were, CX would bisect BD, but it does not (unless ABCD were a rhombus). The correct solution: We are looking for $$\frac{CX*BX}{2}$$ We know: BD is 25 (3-4-5 right triangle * 5), Area of BCD = 150, and therefore CX = 12 (because $$\frac{CX*BD}{2} = 150$$) So therefore: $$BX^2 + 144 = 400$$-> $$BX^2 = 256$$ -> $$BX = 16$$ -> Area is $$\frac{16*12}{2} = 96$$ Hi, Thanks for prompt response, I an't saying angle CXD and angle XCB are 45 digrees, what i am saying angle CXD is 90 and angle CDX are 45 digrees, since DB is diognal of restangle hence in trangle CXD , angle CXD is 90 , CDX is 45 then XCD must be 180-90-45=45, by that CX and DX must be 15/root2, which is not inline with correct explanation Experts please explain Hi vipulgoel, The diagonals of a rectangle don't bisect the side angles. Using the image in the question statement Had diagonal BD bisected ∠B and ∠D, then ∠ABD = ∠ADB = $$\frac{90}{2} = 45$$. As sides opposite equal angels are equal, that would mean AB = AD. If AB = AD, it does not remain a rectangle anymore . It becomes a square. Therefore ∠CDB is not 45 degress. Hope it's clear Regards Harsh _________________ | '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com Manager Joined: 04 Jul 2017 Posts: 59 Location: India Concentration: Marketing, General Management GPA: 1 WE: Analyst (Consulting) Re: If ABCD is a rectangle, what is the area of the shaded region? [#permalink] ### Show Tags 20 Aug 2017, 09:25 SMEs please let me know what's wrong with my approach as I am getting 100 which is not an answer option but close to it. I saw that its a rectangle. Therefore angle CBD has to be 45 since segment BD is a diagonal. Therefore, the shaded region is a 45-45-90 triangle (one angle 90, other 45, so remaining can't be anything but 45). Given that BC is 20 and side opposite to 90 angle is root2*x, x=20/root2=base=height. Solving for 1/2 * b * h = 1/2 * 20 / root2 * 20 / root2=100. Can't understand where am I wrong :( EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 13553 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If ABCD is a rectangle, what is the area of the shaded region? [#permalink] ### Show Tags 20 Aug 2017, 11:26 1 aashishagarwal2 wrote: SMEs please let me know what's wrong with my approach as I am getting 100 which is not an answer option but close to it. I saw that its a rectangle. Therefore angle CBD has to be 45 since segment BD is a diagonal. Therefore, the shaded region is a 45-45-90 triangle (one angle 90, other 45, so remaining can't be anything but 45). Hi aashishagarwal2, The diagonal of a SQUARE will form 45 degree angles; since this is a rectangle with different side lengths, we know that those angles CANNOT be 45 degrees (since that shape is NOT a square). That incorrect assumption is why the rest of your calculation is incorrect. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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20 Aug 2017, 11:53
1

ABCD is rectangle therefore diogonal can be calculated by pythogoras theorm and will be equal to 25
and applying simple trigonometry we can calculate required sides.
and finally area of shaded region
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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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20 Aug 2017, 12:00
chetan2u wrote:
a way different from the one mentioned ....ans C
we can use similar triangles property between triangles abd and bcp and get the sides of triangle bcp and so area comes as 96..

i guess using trigonometry and catching angles 37 and 53 is fastest way, as in my solution..
although trigonometry is not included in gmat , but if one knows , can help him a lot.

Thanks.
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Re: If ABCD is a rectangle, what is the area of the shaded region?  [#permalink]

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20 Oct 2018, 08:09
PareshGmat wrote:

Attachment:
Untitled.png

Diagonal of rectangle $$= \sqrt{20^2 + 15^2} = \sqrt{625} = 25$$

Half the area of rectangle $$= 150 = \frac{1}{2} * 25 * h$$

Height = 12 = CP

Length BP $$= \sqrt{20^2 - 12^2} = \sqrt{256} = 16$$

Area of shaded region $$\triangle BPC = \frac{1}{2} * 16 * 12 = 96$$

How do we know that half of area of rectangle is height ?
Re: If ABCD is a rectangle, what is the area of the shaded region?   [#permalink] 20 Oct 2018, 08:09
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