In the figure above, a right triangle is located entirely inside a squ

Question

https://gmatclub.com/forum/download/file.php?id=26884 In the figure above, a right triangle is located entirely inside a square with side length of 6. If all three side lengths of the right triangle are integers, what fraction of the square is shaded? A. 1/6 B. 1/4 C. \frac{1}{\sqrt{3}} D. \frac{\sqrt{2}}{3} E. It cannot be determined 2015-06-09_1359.png

chetan2u · Answer

Only right angle triplet with base and hyp less than 6 (as the triangle is fully inscribed in square) is 3,4,5..
area of this triangle = 3*4/2=6..
area of square = 6*6=36..
fraction of square covered by triangle=6/36=1/6... ans A

bluesquare · Answer

If the triangle is a right triangle with integer sides, it must be a 3-4-5 triangle, since the next right triangle with integer sides is 5-12-13 and 12 > 6, the size of the square, so the triangle would not fit in the square.

The diagonal of the square is  6\sqrt{2}  since the side is 6, which is greater than 5, so the triangle will fit in the square.

Therefore, the ratio of the shaded region is the  \frac{Area of Triangle}{Area of Square}=\frac{6}{36}=\frac{1}{6}=A

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

The area of the square is (6)(6) = 36. But what is the area of the shaded right triangle?

The constraint that all three sides of the right triangle must be integers is actually quite restrictive. The sides of the triangle must be a Pythagorean triple or a multiple thereof. The first few Pythagorean triples are 3–4–5, 5–12–13, and 8–15–17. Only 3–4–5 is compact enough to fit inside region D, since all other Pythagorean triples have a side longer than 6. We should also check whether ABC could be a multiple of 3–4–5. The smallest integer multiple of the 3–4–5 triple is 6–8–10, which certainly would not fit inside a 6 by 6 square.

Thus, we can conclude that ABC has sides of length 3, 4, and 5, and an area of 1/2*3*4 = 6.

The shaded area is therefore 6/36 = 1/6 of the area of the square.

The correct answer is A.

xLUCAJx · Answer

The wording threw me off a bit and couldn't decide if the 6 was for the triangle or the square, but having all three sides of the triangle be an integer makes it clear.

KrishnakumarKA1 · Answer

Pythagorean triplet with all side less than 6 will be 3,4,5
Area = 1/2 *3*4 = 6
area of square = 6*6 = 36
fraction shaded = 6/36 = 1/6

option A

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In the figure above, a right triangle is located entirely inside a squ

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