In triangle ABC, AB = 5 and AC = 3. which one of the following is the measure of the length of side BC?

Options

A. `BC lt 7`
B. `BC = 7`
C. `BC gt 7`
D. It cannot be determined from the information given

Answer

To find the length of side BC in triangle ABC, where AB = 5 and AC = 3, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

### Step-by-step Solution:

1. **Identify the sides of the triangle:**
- Let AB = 5
- Let AC = 3
- Let BC = x (the side we need to find)

2. **Apply the triangle inequality theorem:**
- According to the triangle inequality, we have the following inequalities:
1. AB + AC > BC
2. AB + BC > AC
3. AC + BC > AB

3. **Substituting the known values:**
- From the first inequality:
\[ 5 + 3 > x \implies 8 > x \implies x < 8 \]
- From the second inequality:
\[ 5 + x > 3 \implies x > 3 - 5 \implies x > -2 \quad (\text{This is always true since lengths are positive}) \]
- From the third inequality:
\[ 3 + x > 5 \implies x > 5 - 3 \implies x > 2 \]

4. **Combining the inequalities:**
- From the inequalities derived, we have:
\[ 2 < x < 8 \]

5. **Conclusion:**
- The length of side BC must be greater than 2 and less than 8. Therefore, BC can take any value in the range (2, 8).