Question

Determine the length of the third side of a triangle from the choices provided if the first two sides measure 10 and 12.(1 point) Responses 24 24 30 30 22 22 15

Answer 1

To determine the possible lengths of the third side of a triangle given the lengths of the other two sides (10 and 12), 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 remaining side.

Let the third side be denoted as \( c \). The triangle inequality gives us the following three conditions:

1. \( 10 + 12 > c \)
2. \( 10 + c > 12 \)
3. \( 12 + c > 10 \)

Now, let's evaluate each condition:

1. From \( 10 + 12 > c \), we get:
\[
22 > c \quad \text{or} \quad c < 22
\]

2. From \( 10 + c > 12 \), we get:
\[
c > 2
\]

3. From \( 12 + c > 10 \), we get:
\[
c > -2 \quad \text{(which is always true if \( c > 2 \))}
\]

Combining the results from the inequalities, we find:
\[
2 < c < 22
\]

Now we check the provided choices: 24, 30, 22, and 15.

- 24: Not valid (greater than 22)
- 30: Not valid (greater than 22)
- 22: Not valid (not less than 22)
- 15: Valid (falls within the range 2 < c < 22)

Thus, the only suitable length for the third side of the triangle is **15**.