To determine the number of integer values of x for which the triangle will be an obtuse angled triangle, we need to use the concept of the triangle inequality theorem and the Pythagorean theorem.

1. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have sides of lengths 15 cm, 8 cm, and x cm. So, we can write the following inequalities:

a) 15 + 8 x
b) 15 + x 8
c) 8 + x 15

2. Simplifying these inequalities, we get:

a) 23 x
b) 15 x
c) x 7

3. Now, let's consider the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, we want to find the values of x for which the triangle is obtuse, meaning that the square of the longest side is greater than the sum of the squares of the other two sides.

Using the Pythagorean theorem, we can write the following inequality:

d) x^2 15^2 + 8^2

4. Simplifying this inequality, we get:

d) x^2 289

5. Taking the square root of both sides, we have:

e) x √289

6. Simplifying further, we get:

e) x 17

7. Combining the inequalities from steps 2 and 6, we find that the values of x must satisfy the following conditions:

23 x 17

8. To find the number of integer values of x within this range, we subtract the lower limit from the upper limit and add 1:

Number of integer values of x = 23 - 17 + 1 = 7

Therefore, there are 7 integer values of x for which the triangle will be an obtuse angled triangle.

Question

To determine the number of integer values of x for which the triangle will be an obtuse angled triangle, we need to use the concept of the triangle inequality theorem and the Pythagorean theorem.

1. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we have sides of lengths 15 cm, 8 cm, and x cm. So, we can write the following inequalities:

a) 15 + 8 > x
   b) 15 + x > 8
   c) 8 + x > 15

2. Simplifying these inequalities, we get:

a) 23 > x
   b) 15 > x
   c) x > 7

3. Now, let's consider the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, we want to find the values of x for which the triangle is obtuse, meaning that the square of the longest side is greater than the sum of the squares of the other two sides.

Using the Pythagorean theorem, we can write the following inequality:

d) x^2 > 15^2 + 8^2

4. Simplifying this inequality, we get:

d) x^2 > 289

5. Taking the square root of both sides, we have:

e) x > √289

6. Simplifying further, we get:

e) x > 17

7. Combining the inequalities from steps 2 and 6, we find that the values of x must satisfy the following conditions:

23 > x > 17

8. To find the number of integer values of x within this range, we subtract the lower limit from the upper limit and add 1:

Number of integer values of x = 23 - 17 + 1 = 7

Therefore, there are 7 integer values of x for which the triangle will be an obtuse angled triangle.

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