How many integers values of x are there so that x, 5, and 8 could be the lengths of the sides of a triangle?
Please do not add second questions to your own question thread. It is confusing for the teachers.
The possible side lengths in your first question are between 8-5 (=3) and 8+5(=13) . Allowed integer values would be 4,5,6,7,8,9,10, 11 and 12.
That makes 9 values.
In your second question, the third angle of the triangle must be 18 degrees. The shortest side is opposite the smallest angle (A).
In ΔABC, m∠C = 118 and m∠B = 44. Which is the shortest side of the triangle?
To determine the number of integer values of x that could be the lengths of the sides of a triangle with side lengths x, 5, and 8, we need to consider the triangle inequality theorem.
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.
Therefore, we need to check the following three inequalities:
1. x + 5 > 8 (length of the third side)
2. x + 8 > 5 (length of the second side)
3. 5 + 8 > x (length of the first side)
Simplifying each inequality, we have:
1. x > 3
2. x > -3
3. 13 > x
Now, we need to find the values of x that satisfy all three inequalities simultaneously:
Since x must be greater than 3 and greater than -3, the only possible values are x = 4, 5, 6, ..., 12.
Additionally, x must be less than 13. Therefore, the valid values of x are x = 4, 5, 6, ..., 12.
Counting these values, we find that there are 12 valid integer values of x.
To determine the number of integer values of x that satisfy the given condition, we need to apply the triangle inequality theorem. According to this 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, let's consider x as the length of one side of the triangle, while 5 and 8 represent the other two sides.
For x, 5, and 8 to be the lengths of a triangle, we need to satisfy three conditions:
1. x + 5 > 8
2. 5 + 8 > x
3. 8 + x > 5
To determine the integer values of x that satisfy these conditions, we can solve each inequality separately.
1. x + 5 > 8:
Subtracting 5 from both sides: x > 3
2. 5 + 8 > x:
Combining the terms: 13 > x
3. 8 + x > 5:
Subtracting 8 from both sides: x > -3
Now, let's combine all the conditions:
3 < x < 13
Therefore, there are 9 integer values of x (4, 5, 6, 7, 8, 9, 10, 11, 12) that satisfy the given condition.