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Geometry and Trigonometry Difficulty: Hard

The figure presents right triangle A B C, where angle C is a right angle. Side A C is horizontal, such that vertex A is to the left of vertex C, and vertex B is directly above vertex C. A vertical line segment is drawn from a point on side A B to a point on side A C and divides side A C into two segments. The length of the segment between vertex A and the vertical line segment is labeled 5 and the length of the segment between the vertical line segment and vertex C is labeled 7. The vertical line segment is labeled x. Side B C is labeled y. A note states the figure is not drawn to scale.

The area of triangle ABC above is at least 48 but no more than 60. If y is an integer, what is one possible value of x ?

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Explanation

The correct answer is either $the fraction 10 over 3$ , $the fraction 15 over 4$ , or $the fraction 25 over 6$ . The area of triangle ABC can be expressed as $one half times, open parenthesis, 5 plus 7, close parenthesis, times y$ or $6 y$ . It’s given that the area of triangle ABC is at least 48 but no more than 60. It follows that $48 is less than or equal to 6 y, which is less than or equal to 60$ . Dividing by 6 to isolate y in this compound inequality yields $8 is less than or equal to y, which is less than or equal to 10$ . Since y is an integer, $y equals 8, 9, or 10$ . In the given figure, the two right triangles shown are similar because they have two pairs of congruent angles: their respective right angles and angle A. Therefore, the following proportion is true: $the fraction x over y equals the fraction 5 over 12$ . Substituting 8 for y in the proportion results in $the fraction x over 8 equals the fraction 5 over 12$ . Cross multiplying and solving for x yields $the fraction 10 over 3$ . Substituting 9 for y in the proportion results in $the fraction x over 9 equals the fraction 5 over 12$ . Cross multiplying and solving for x yields $the fraction 15 over 4$ . Substituting 10 for y in the proportion results in $the fraction x over 10 equals the fraction 5 over 12$ . Cross multiplying and solving for x yields $the fraction 25 over 6$ . Note that 10/3, 15/4, 25/6, 3.333, 3.75, 4.166, and 4.167 are examples of ways to enter a correct answer.