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The GMAT Math practice question is a geometry problem to find the number of acute angled triangles that are possible given two sides of a triangle.

If 10, 12 and 'x' are sides of an acute angled triangle, how many integer values of 'x' are possible?

1. 7
2. 12
3. 9
4. 13
5. 11

The correct choice is (C) and the correct answer is 9 values.

Finding the answer to this question requires one to know two rules in geometry.

Rule 1: For an acute angled triangle, the square of the LONGEST side MUST BE LESS than the sum of squares of the other two sides.

Rule 2: For any triangle, sum of any two sides must be greater than the third side.

The sides are 10, 12 and 'x'.

From Rule 2, x can take the following values: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 – A total of 19 values.

When x = 3 or x = 4 or x = 5 or x = 6, the triangle is an OBTUSE angled triangle (Rule 1 is NOT satisfied).

The smallest value of x that satisfies BOTH conditions is 7. (10² + 7² > 12²).

The highest value of x that satisfies BOTH conditions is 15. (10² + 12² > 15²).

When x = 16 or x = 17 or x = 18 or x = 19 or x = 20 or x = 21, the triangle is an OBTUSE angled triangle (Rule 1 is NOT satisfied).

Hence, the values of x that satisfy both the rules are x = 7, 8, 9, 10, 11, 12, 13, 14, 15. A total of 9 values.

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