If 10, 12 and ‘x’ are Sides of an Acute Angled Triangle, How Many Integer Values of ‘x’ are Possible? GMAT Problem Solving

Question: 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

‘If 20 Men or 24 Women or 40 Boys can do a Job in 12 Days GMAT Problem Solving’ is the topic from the GMAT Quantitative problem set. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

Explanation:

The question asks for an acute triangle i.e. a triangle with all angles less than 90. An obtuse angled triangle has one angle more than 90. So the logic is that before one of the angles reaches 90, find out all the values that x can take.

Starting from the first diagram where x is minimum and the angle is very close to 90, to the 2nd diagram where all angles are much less than 90. Also, in the third diagram the other angle is going towards 90.

Note: The remaining angle cannot be 90 because that will make 10 the hypotenuse but hypotenuse is always the greatest side.

In the leftmost diagram x = \(\sqrt{(12^2)}-\sqrt{(10^2)}\)

x = root 44 which is 6.something
x should be greater than 6. Something because the angle cannot be 90.

In the rightmost diagram, x = \(\sqrt{(12^2)}+\sqrt{(10^2)}\)

x = root 244 which is 15.something
x should be less than 15. Something so that the angle is not 90.

Values that x can take range from 7 to 15 which is

=15 - 7
= 8 + 1
= 9 values.

Hence, C is the correct answer.

Approach Solution 2:

Explanation:

The catch in the acute angled triangle.

We will consider the figures attached...

Case 1:
x

Case 2:
x6

Therefore x ranges from 7 to 15 = 9 Values
Hence, C is the correct answer.

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