Question

Question content area top
Part 1
A triangle has two side lengths of 11.5 inches and 12.75 inches. Which of the following side lengths could be the third​ side?
Question content area bottom
Part 1
Choose the correct answer below.
A.
0.75 in.
B.
1.25 in.
C.
24.25 in.
D.
23.75 in.

Answer 1

To determine the possible lengths of the third side of a triangle given two sides, we can use the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \):

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

In this case, we have two side lengths:
- \( a = 11.5 \) inches
- \( b = 12.75 \) inches

Let's denote the length of the third side as \( c \).

### Applying the Triangle Inequality:

1. From \( a + b > c \):
\[
11.5 + 12.75 > c \implies 24.25 > c \implies c < 24.25
\]

2. From \( a + c > b \):
\[
11.5 + c > 12.75 \implies c > 12.75 - 11.5 \implies c > 1.25
\]

3. From \( b + c > a \):
\[
12.75 + c > 11.5 \implies c > 11.5 - 12.75 \implies c > -1.25 \quad \text{(this doesn't give us further information)}
\]

### Conclusion:

Combining the inequalities from the first two conditions, we have:
\[
1.25 < c < 24.25
\]

Now let's evaluate the given answer choices:

A. \( 0.75 \) in. — **Not valid** (because \( 0.75 < 1.25 \))

B. \( 1.25 \) in. — **Not valid** (because \( c \) must be greater than \( 1.25 \))

C. \( 24.25 \) in. — **Not valid** (because \( c \) must be less than \( 24.25 \))

D. \( 23.75 \) in. — **Valid** (because \( 1.25 < 23.75 < 24.25 \))

Thus, the correct answer is:

**D. 23.75 in.**