Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI.

Question

https://gmatclub.com/forum/download/file.php?id=53888 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 v6Ar1Ro.png

BrentGMATPrepNow · Accepted Answer

Important: Since each right triangle has  integer  lengths, we're looking for  Pythagorean triples  that satisfy the given information. 
 Aside: Pythagorean triples are sets of three integers that could be the measurements of a right triangle.

Some Pythagorean triples include: 
3-4-5
5-12-13
7-24-25
8-15-17
etc

Also note that we can take the above Pythagorean triples and create additional Pythagorean triples by multiplying all values by the same integer. 
For example, we can take the Pythagorean triple 3-4-5, and multiply of the three values by various integers to create additional triples such as:
6-8-10
9-12-15
12-16-20
15-20-25
etc

Notice that the largest number in a triple will be the length of the hypotenuse. So the first two values must be the lengths of the two legs of the right triangle.

Okay, let's first find a Pythagorean triple for the blue right triangle. 
 https://i.imgur.com/eHAR2C8.png  
 Area of triangle = (base)(height)/2 
So, if we let x and y be the lengths of the two legs, we can write:   xy/2 = 96

Now let's find a Pythagorean triple that meets the condition that   xy/2 = 96  
Well, if we have a 12-16-20 right triangle, then the area = (12)(16)/2 = 96. PERFECT!
So the blue right triangle looks like this
 https://i.imgur.com/JjHgTry.png

Now let's first find a Pythagorean triple for the red right triangle. 
If we let j and k be the lengths of the two legs, we can write:   jk/2 = 150  
Among the various Pythagorean triples, we can see that, if we have a 15-20-25 right triangle, then the area = (15)(20)/2 = 150. PERFECT!
So the red right triangle looks like this
 https://i.imgur.com/2esCxZ5.png

When we add the relevant information to our diagram we get: 
 https://i.imgur.com/IMNUYfS.png

We know that:  (area of small rectangle FGHI) = (area of big rectangle ABCD) - (area of the 4 triangles) 
Substitute values to get: area of small rectangle FGHI = (20)(25) - (  150   +   150   +   96   +   96  )
= 500 - 492
= 8

Answer: A

Cheers,
Brent

nick1816 · Answer

As all the sides of the triangles are integers, we have to look for Pythagorean triplets

96= 1*96 = 2*48=3*32=4*24= 6*16= 8*12

Length of sides of triangle AID are 12,16 and 20 cm.

150=1*150 = 2*75=3*50= 5*30=6*25=  10*15

Length of sides of triangle DHC are 15,20 and 25 cm.

Area of ABCD = 20*25=500
Area of 4 triangles = 150+150+96+96= 492

Area of FGHI = 500-492 = 8

OR

As AF is smaller than one of the legs of triangle AID, AF = 15 and AI =16. Hence, FI = 16-15=1 cm

Dh =20 and DI=12; Hence HI =8cm

Area of FGHI = 1*8=8 cm^2

Kinshook · Answer

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

IMO A

Kritisood · Answer

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?

nick1816 · Answer

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.

Posted from my mobile device

BrentGMATPrepNow · Answer

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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Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI.

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Rectangle ABCD is comprised of 4 right triangles and rectangle FGHI.

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