ardz24 wrote:Consider a triangle with sides 9 cm, 16 cm and x cm. How many such triangles exist?
(1) The triangle is obtuse-angled triangle
(2) x is an integer
What's the best way to determine which statement is sufficient? Can any experts assist?
Note the two rules for a triangle.
1. The sum of any two sides is greater than the third side. Thus, 9 + 16 > x => 25 > x.
2. The difference between any two sides is less than the third side. Thus, 16 - 9 < x => 7 < x.
=> 7 < x < 25
Let's see each statement one by one.
(1) The triangle is an obtuse-angled triangle.
There can be an uncountable number of obtuse-angled triangles, considering x as the greatest side and x as the greatest side. Note that we need not consider that x is an integer. For example, a triangle with sides 9, 9, and 16 is different than a triangle with sides 9.1, 9, and 16. Insufficient.
(2) x is an integer
Considering 7 < x < 25, x can be 8, 9, 10, ... 24; a total of 17 triangles. But we do not know whether the triangle is an acute-angled, rightangled, or an obtuse-angled triangle. Accordingly, the number of triangles will vary. Insufficient.
(1) and (2) together
Case 1: x is the greatest side.
=> 9^2 + 16^2 < x^2 => 81 + 256 < x^2 => 337 < x^2 => 324 < 337 < 361 => 18^2 < 337 < 19^2 => 18 < x. The possible values of x are 19, 20, 21, 22, 23, and 24. There are six possible obtuse-angles trianles.
Case 2: 16 is the greatest side.
=> 9^2 + x^2 < 16^2 => 81 + x^2 < 256 => x^2 < 256 - 81 => x^2 < 175 => x^2 < 196 => x^ < 14^2 => x < 14. The possible values of x are 8, 9, 10, 11, 12, and 13. There are six possible obtuse-angles trianles.
Each possible case renders six possible obtuse-angles trianles. Sufficient.
The correct answer:
C
Hope this helps!
-Jay
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