The sides of a triangle are 6.5 cm, 10 cm and x cm, where x is a positive number. What is the smallest possible value of x among the following.

Options

A. 3.5
B. 4
C. 4.5
D. 2.8

Answer

To find the smallest possible value of \( x \) for the triangle with sides 6.5 cm, 10 cm, and \( x \) cm, we will use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We will apply this theorem to our triangle.

### Step 1: Apply the Triangle Inequality Theorem
We have three sides: 6.5 cm, 10 cm, and \( x \) cm. We need to set up inequalities based on the triangle inequality theorem.

1. \( 6.5 + 10 > x \)
2. \( 6.5 + x > 10 \)
3. \( 10 + x > 6.5 \)

### Step 2: Solve the Inequalities

**Inequality 1:**
\[ 6.5 + 10 > x \]
\[ 16.5 > x \]
This means:
\[ x < 16.5 \]

**Inequality 2:**
\[ 6.5 + x > 10 \]
\[ x > 10 - 6.5 \]
\[ x > 3.5 \]

**Inequality 3:**
\[ 10 + x > 6.5 \]
\[ x > 6.5 - 10 \]
\[ x > -3.5 \]
Since \( x \) is a positive number, this inequality does not provide a new constraint.

### Step 3: Combine the Results
From the inequalities we derived:
- From Inequality 1, we have \( x < 16.5 \).
- From Inequality 2, we have \( x > 3.5 \).

Thus, we can conclude:
\[ 3.5 < x < 16.5 \]

### Step 4: Determine the Smallest Possible Value of \( x \)
The smallest possible value of \( x \) that satisfies the inequality \( x > 3.5 \) is just above 3.5. However, since we are looking for a specific value among given options, we would choose the smallest option that is greater than 3.5.

### Conclusion
The smallest possible value of \( x \) is any value greater than 3.5. If options are provided, select the smallest one that is greater than 3.5.

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