Geometry concepts such as basic properties of triangles and how to find if a given triangle is a right triangle or an acute angled triangle or an obtuse angled triangle are often tested in GMAT.
Here is a question using two basic rules about triangles.
If 10, 12 and 'x' are sides of an acute angled triangle, how many integer values of 'x' are possible?
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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Labels: acute angled triangles, GMAT Geometry, triangles