Question

Question
Apply the Triangle Inequality Theorem to determine the possible whole number measures of the third side of a triangle if the first two sides measure 6 and 2. List them in ascending order.(2 points)
The measure of the third side could be
,
, or
.

Answer 1

most certain

Answer 2

absolutely positive

Answer 3

To apply the Triangle Inequality Theorem, we need to make sure that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Let's denote the first two sides as side A with a length of 6 and side B with a length of 2.

Using the Triangle Inequality Theorem, we can write the following inequalities:

1. A + B > C
2. B + C > A
3. A + C > B

Substituting the known values:

1. 6 + 2 > C
2. 2 + C > 6
3. 6 + C > 2

Simplifying the inequalities:

1. 8 > C
2. C + 2 > 6
3. C + 6 > 2

From the first inequality, we know that C must be less than 8. From the second inequality, we know that C must be greater than 4. From the third inequality, we know that C must be greater than -4.

Combining all the above conditions:

4 < C < 8

Since we are looking for whole number measures of the third side, the possible integer values for C within the given range are: 5, 6, and 7.

Listing them in ascending order: 5, 6, 7

Answer 4

The most certain whole number measures of the third side of the triangle are 5, 6, and 7.

Answer 5

I am absolutely positive that the only possible whole number measures of the third side of the triangle are 5, 6, and 7.