Question: Consider an obtuse-angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer then how many such triangles exist?

1. 5
2. 21
3. 10
4. 15
5. 14

Correct Answer: (C)

Solutions and Explanation

Approach Solution 1 :

This right-angled triangle has sides of 6, 8, and 10, and a hypotenuse of 10. Let the sides be 6, 8, and 10.
6,8,10 cannot be sides of an obtuse angled triangle because no angle is greater than 90.

Case - 1 : The largest side is 15.

If the angle is 90 degrees, the x value will be √[(15^2) - (8^2)] = √161. This will be between 12 and 13.
If we take 13, the angle will be less than 90 degrees, so 12 and the values below it are correct.

However, the minimum value is 15-8 +1= 7+1 = 8, making the values from 8 to 12 making a number of 5 values.

Case - 2 : x is the largest side;

Likewise for it to be 90 degree, x must be √[(15^2) + (8^2)] = 17
So it must be greater than 17.
The maximum values will be, 5 + 18 -1 = 22
Therefore the Values are 18, 19, 20, 21, and 22 making a number of 5 values.

As a result, 5+5=10 values in total.

Approach Solution  2 :

There must be an angle that is greater than 90 degrees.
a^2 + b^2 < c^2

The side opposite of the obtuse angle is z, and it is consequently the longest.
If the biggest side is 15, then:
=> 8^2 + x^2 < 15^2
= 64 + x^2 < 225
= x^2 < 161

In accordance with our other range, x could therefore be either 8, 9, 10, 11, or 12.

If x is our large side, then we have:
=> 8^2 + 15^2 < x^2
= 64 + 225 < x^2
= 289 < x^2
= 17 < x

x might be 18 or 19 or 20 or 21 or 22.
Therefore x has 10 possible values.

Approach Solution 3 :

There are two theories that can be considered here.

1. Pythagoras is one example of an acute triangle. Here c is the largest side of the three, -c2>a2+b2.

2. Theorem of triangle inequality. A triangle's two sides must have a sum greater than its third side's measurement. c, which is the largest side of the three, equals a plus b.

Let x be the bigger side :
All possible values of x, according to the triangle inequality theorem, are ||22,21,20,19,18,17,16,15||.
However, if x=15, 16, or 17, c2>a2+b2 is not satisfied, so according to Pythagoras, we can only take values higher than 17. Which implies that x=22,21,20,19,18.
Let 15 be the bigger side :
The triangle inequality theorem states that the possible values of "x" are 8, 9, 10, 11, 12, 13, and 14.
However, if x=14 or 13, c2>a2+b2 is not satisfied, so according to Pythagoras, we can only take into account values lower than 13. Which implies that x=8,9,10,11,12
As a result, all possible values x can be 8, 9, 10, 11, 12, 18, 19, 20, 21, and 22.
There are a total of ten values.

“Consider an obtuse-angled triangles with sides 8 cm, 15 cm” - is a topic that is covered in the quantitative reasoning section of the GMAT. To successfully execute the GMAT Problem Solving questions, a student must possess a wide range of qualitative skills. The entire GMAT Quant section consists of 31 questions. The problem-solving section of the GMAT Quant topics requires the solution of calculative mathematical problems.

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