Question

A triangle has two side lengths of 10.5 inches and 15.75 inches. Which of the following side lengths could be the third​ side?
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Part 1
Choose the correct answer below.
A.
4.75 in.
B.
5.25 in.
C.
25.75 in.
D.
26.25 in.

Answer 1

To find the possible lengths of the third side of a triangle when the two sides are known, we use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let the side lengths be \( a = 10.5 \) inches, \( b = 15.75 \) inches, and \( c \) be the length of the third side we want to determine.

The three inequalities derived from the triangle inequality theorem are as follows:

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

Using the given values, we can evaluate each inequality:

1. **First inequality**:
\[
10.5 + 15.75 > c \\
26.25 > c \\
c < 26.25
\]

2. **Second inequality**:
\[
10.5 + c > 15.75 \\
c > 15.75 - 10.5 \\
c > 5.25
\]

3. **Third inequality**:
\[
15.75 + c > 10.5 \\
c > 10.5 - 15.75 \\
c > -5.25 \quad (\text{this is always true for positive } c)
\]

Combining the results of the first two inequalities, we get:
\[
5.25 < c < 26.25
\]

Now we can check which of the options provided fit this range:

A. \( 4.75 \) in. ⟹ **Not valid** (since \( 4.75 < 5.25 \))

B. \( 5.25 \) in. ⟹ **Not valid** (since \( c \) must be greater than \( 5.25 \))

C. \( 25.75 \) in. ⟹ **Valid** (since \( 25.75 < 26.25 \))

D. \( 26.25 \) in. ⟹ **Not valid** (since \( c \) must be less than \( 26.25 \))

Based on this analysis, the only valid option for the length of the third side is:

**C. 25.75 in.**