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

2 mm
2 mm

1 mm
1 mm

20 mm

Answer 1

To determine the possible length of the third side of a triangle when two sides are 10 mm and 7 mm, 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.

Let's denote the lengths of the sides as follows:
- \( a = 10 \) mm
- \( b = 7 \) mm
- \( c \) = length of the third side

According to the triangle inequality theorem, we have the following inequalities:

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

Substituting the known values:

1. \( 10 + 7 > c \) ⟹ \( 17 > c \) ⟹ \( c < 17 \)
2. \( 10 + c > 7 \) ⟹ \( c > -3 \) (which doesn't restrict the possible value since lengths are positive)
3. \( 7 + c > 10 \) ⟹ \( c > 3 \)

Combining the inequalities that restrict \( c \):
- From 1: \( c < 17 \)
- From 3: \( c > 3 \)

Thus, the possible range for \( c \) is \( 3 < c < 17 \).

Looking at the options given:
- 5 mm
- 2 mm
- 1 mm
- 20 mm

The only value that lies within the range \( 3 < c < 17 \) is **5 mm**.

So, the length of the third side must be **5 mm**.