Question

two sides of a triangle mesure 6in, and 8in. what are possible lengths of the third side?
1in
2in
4in
10in
12in
14in
15in

Answer 1

To determine the possible lengths of the third side of a triangle when the other two sides measure 6 inches and 8 inches, you can apply 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 the lengths of the two known sides be \( a = 6 \) inches and \( b = 8 \) inches, and let \( c \) be the length of the third side. According to the triangle inequality theorem, the following inequalities must hold:

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

Substituting the known values into these inequalities:

1. \( 6 + 8 > c \) → \( 14 > c \) → \( c < 14 \)
2. \( 6 + c > 8 \) → \( c > 2 \)
3. \( 8 + c > 6 \) → \( c > -2 \) (this is always true since \( c \) must be positive)

From the first two inequalities, we find:
- \( c < 14 \)
- \( c > 2 \)

Combining these inequalities, we have:

\[
2 < c < 14
\]

Therefore, the possible integer lengths for \( c \) based on the provided options are:

- 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

From the options given (1in, 2in, 4in, 10in, 12in, 14in, 15in), the possible lengths of the third side that satisfy the triangle inequality are:

- **4 inches**
- **10 inches**
- **12 inches**

Thus, the possible lengths of the third side are **4 inches, 10 inches, and 12 inches**.