in triangle ABC, AB=6, BC=12 AND AC=X. Which of the following cannot be a value of x?
a. 6
b.7
c.8
d.9
e.10
To determine which value cannot be a value of x in triangle ABC, we need to apply 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.
Let's analyze each option:
a. If AC = 6, then the sum of AB and BC would be 6 + 12 = 18, which is greater than AC. Therefore, option a is a valid value for x.
b. If AC = 7, then the sum of AB and BC would be 6 + 12 = 18, which is greater than AC. Therefore, option b is a valid value for x.
c. If AC = 8, then the sum of AB and BC would be 6 + 12 = 18, which is equal to AC. In this case, the triangle would be degenerate, meaning the three points A, B, and C would all lie on the same line and not form a triangle. Therefore, option c cannot be a value for x.
d. If AC = 9, then the sum of AB and BC would be 6 + 12 = 18, which is greater than AC. Therefore, option d is a valid value for x.
e. If AC = 10, then the sum of AB and BC would be 6 + 12 = 18, which is equal to AC. Similar to option c, the triangle would be degenerate, so option e cannot be a value for x.
Therefore, the value that cannot be a value of x is option c.
To determine which value of x cannot be the length of AC in triangle ABC, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Using this rule, we can compare the lengths of the sides:
AC + AB > BC
AC + 6 > 12
AC > 6
Therefore, the length of AC must be greater than 6.
Now let's check each option:
a. If AC = 6, this is not possible as AC must be greater than 6. Therefore, option a is not a valid value for x.
b. If AC = 7, this is possible as AC is greater than 6. Therefore, option b is a valid value for x.
c. If AC = 8, this is possible as AC is greater than 6. Therefore, option c is a valid value for x.
d. If AC = 9, this is possible as AC is greater than 6. Therefore, option d is a valid value for x.
e. If AC = 10, this is possible as AC is greater than 6. Therefore, option e is a valid value for x.
Therefore, the answer is option a. AC cannot be equal to 6.
each side must be less than the sum of the other two sides, so we need
x < 6+12
6 < x+12
12 < x+6
looks like (a) to me. If the sides are 6,6,12, then the triangle is just a flat line.