Question

Two sides of a triangle measure 5 in. and 12 in. Which could be the length of the third side?
3 in.
6 in.
10 in.
18 in.

Answer 1

To determine the possible lengths of the third side of a triangle given two sides of lengths \( a = 5 \) in. and \( b = 12 \) in., we can use the triangle inequality theorem. The triangle inequality 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 third side as \( c \). The triangle inequalities can be expressed as:

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

Substituting the known values \( a = 5 \) in. and \( b = 12 \) in., we get:

1. \( 5 + 12 > c \) → \( 17 > c \) → \( c < 17 \)
2. \( 5 + c > 12 \) → \( c > 7 \)
3. \( 12 + c > 5 \) → this inequality will always be satisfied for positive \( c \) so it does not provide a constraint.

Combining the inequalities from 1 and 2, we have:

- \( 7 < c < 17 \)

Now, we can evaluate the provided options:

- 3 in.: This does not satisfy \( 7 < c \).
- 6 in.: This does not satisfy \( 7 < c \).
- 10 in.: This satisfies \( 7 < 10 < 17 \).
- 18 in.: This does not satisfy \( c < 17 \).

Thus, the length of the third side could be **10 in.**