Question

Two sides of a triangle are 10 mm and 7 mm. Determine the length of the third side from the choices given.(1 point) Responses 5 mm 5 mm 20 mm 20 mm 1 mm 1 mm 2 mm

Answer 1

To determine the possible lengths of the third side of a triangle given two sides, we can use the triangle inequality theorem. The 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.

Let \( a = 10 \, \text{mm} \) and \( b = 7 \, \text{mm} \). Let \( c \) be the length of the third side. According to the triangle inequality theorem, we have:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Now, let's apply these inequalities:

1. **From \( a + b > c \)**:
\[
10 + 7 > c \implies 17 > c \implies c < 17
\]

2. **From \( a + c > b \)**:
\[
10 + c > 7 \implies c > 7 - 10 \implies c > -3 \quad \text{(this is always true)}
\]

3. **From \( b + c > a \)**:
\[
7 + c > 10 \implies c > 10 - 7 \implies c > 3
\]

Combining these results:
- From the inequalities, we have \( 3 < c < 17 \).

Now, let's examine the choices provided:
- 5 mm (valid)
- 20 mm (invalid, as \( c < 17 \))
- 1 mm (invalid, as \( c > 3 \))
- 2 mm (invalid, as \( c > 3 \))

The only valid option according to the triangle inequalities is **5 mm**.

Thus, the length of the third side can be **5 mm**.