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
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.
Area of triangle = (base)(height)/2So, if we let x and y be the lengths of the two legs, we can write:
xy/2 = 96Now let's find a Pythagorean triple that meets the condition that
xy/2 = 96Well, 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
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 = 150Among 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
When we add the relevant information to our diagram we get:
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