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09 Jun 2015, 03:00
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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
Attachment:
2015-06-09_1359.png [ 15.51 KiB | Viewed 7646 times ]
09 Jun 2015, 03:08
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Bunuel
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
10 Jun 2015, 01:41
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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 \(\frac{Area of Triangle}{Area of Square}=\frac{6}{36}=\frac{1}{6}=A\)
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 \(\frac{Area of Triangle}{Area of Square}=\frac{6}{36}=\frac{1}{6}=A\)
