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Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI. [#permalink]
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BrentGMATPrepNow wrote:


Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI. Triangles ADI and CBG are identical, and each has an area of 96. Triangles AFB and CHD are identical, and each has an area of 150. If the sides of each triangle have integer lengths, what is the area of rectangle FGHI?

A) 8
B) 10
C) 12
D) 14
E) 16


Attachment:
v6Ar1Ro.png


AI * ID = GB * GC = 96*2 = 192 = 2*2*2*2*2*2*3 = 2^6*3 = 8*24 = 16*12
HD* HC = FA * FB = 300 = 2^2*3*5^2 = 20*15

Let us assume AI = 16, ID = 12; AF = 15, FB = 20
AI^2 + ID^2 = 16^2 + 12^2 = 20^2 = AD^2; AD=20 = BC
AF^2 + FB^2 = 15^2 + 20^2 = 25^2 = AB^2 ; AB=25= CD

FI = AI - AF = 16 - 15 = 1
FG = FB - GB = 20 - 12 = 8

Area FGHI = 1*8 = 8

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Re: Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI. [#permalink]
Hi, I have a query on the application of Pythagorean triplets. Just cus a triangle is a right-angled triangle doesn't mean it would be 3-4-5 triplet. It can take values of any of the triplets..so in this question how are we assuming certain triplets with surety? I mean they can easily take other values as well, right?
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Re: Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI. [#permalink]
96= 1*96 = 2*48=3*32=4*24=6*16= 8*12

Take first case 1*96

Nearest square of 96 is (96+95) or (96+97) away. You can easily eliminate this case. As adding square of 2 in 96^2 won't give you a perfect square. Similarly you can eliminate 2nd and 3rd case.

8 and 24 are not co-prime. So deduce their ratio to the lowest form, that is 1:3. 1^2+3^2 is not a perfect square. 4 and 48, or 1 and 12 again won't give you Pythagorean triplet.( Explanation is similar to above cases)

Now you left with last 2 cases. Both will give you ( 12,16,20) triplet.

Kritisood wrote:
Hi, I have a query on the application of Pythagorean triplets. Just cus a triangle is a right-angled triangle doesn't mean it would be 3-4-5 triplet. It can take values of any of the triplets..so in this question how are we assuming certain triplets with surety? I mean they can easily take other values as well, right?


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Re: Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI. [#permalink]
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Kritisood wrote:
Hi, I have a query on the application of Pythagorean triplets. Just cus a triangle is a right-angled triangle doesn't mean it would be 3-4-5 triplet. It can take values of any of the triplets..so in this question how are we assuming certain triplets with surety? I mean they can easily take other values as well, right?


Fortunately there aren't many Pythagorean triples to choose from (given the relatively small triangle areas)

Also notice that, since 96 = (2)(2)(2)(2)(2)(3), the lengths of the two legs must be multiples of 2 and 3 only.
So, for example, this rules out any version of the 5-12-13 triangle and the 7-24-25 triangle.

Also, once we recognize that the red and blue triangles are both similar triangles.
So, once we know that the blue triangles have dimensions 12-16-20 (a version of the 3-4-5 triangle), we can also conclude that the blue triangles are also versions of the 3-4-5 right triangle.

Cheers,
Brent
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Re: Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI. [#permalink]
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Re: Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI. [#permalink]
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