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

20 mm
20 mm

1 mm
1 mm

2 mm
2 mm

5 mm

Answer 1

To determine the possible length of the third side of a triangle when the other two sides are known, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the lengths of the two sides be \( a = 10 \) mm and \( b = 7 \) mm. Let's denote the length of the third side as \( c \).

The triangle inequalities can be expressed as:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Substituting \( a \) and \( b \):

1. \( 10 + 7 > c \) → \( 17 > c \) → \( c < 17 \)
2. \( 10 + c > 7 \) → \( c > -3 \) (this doesn't impose a limit since \( c \) must be positive)
3. \( 7 + c > 10 \) → \( c > 3 \)

Combining these inequalities, we have:
\( 3 < c < 17 \)

Now, let's check the provided choices:
- 20 mm: Does not satisfy \( c < 17 \)
- 1 mm: Does not satisfy \( c > 3 \)
- 2 mm: Does not satisfy \( c > 3 \)
- 5 mm: Satisfies \( 3 < c < 17 \)

Thus, the only valid option for the length of the third side is **5 mm**.