A triangle has two sides measuring 911 and 13 centimeters. What is the smallest possible perimeter considering that the third side must be an integer?

To solve this problem, we can use the triangle inequality theorem, which states that for any triangle with sides \( a \), \( b \), and \( c \), the following must be true:

\[
a + b > c
\]
\[
a + c > b
\]
\[
b + c > a
\]

Given two sides of the triangle, \( a = 911 \) cm and \( b = 13 \) cm, and the third side \( c \) being an integer, we need to determine the smallest possible perimeter of the triangle.

### Step 1: Apply the triangle inequality

#### Inequality 1: \( a + b > c \)
\[
911 + 13 > c \quad \Rightarrow \quad c < 924
\]
So, \( c \) must be less than 924.

#### Inequality 2: \( a + c > b \)
\[
911 + c > 13 \quad \Rightarrow \quad c > 13 - 911 \quad \Rightarrow \quad c > -898
\]
Since \( c \) must be positive, this condition doesn't give any new restrictions. We already know \( c \) is positive.

#### Inequality 3: \( b + c > a \)
\[
13 + c > 911 \quad \Rightarrow \quad c > 898
\]

### Step 2: Combine the results
From the inequalities, we know:
\[
898 < c < 924
\]
So, the smallest possible integer value for \( c \) is 899.

### Step 3: Calculate the perimeter
The perimeter \( P \) is the sum of all three sides:
\[
P = a + b + c = 911 + 13 + 899 = 1823
\]

### Final Answer:
The smallest possible perimeter is **1823 cm**.

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. How would the problem change if one of the given sides was not an integer?
2. What is the largest possible perimeter for this triangle?
3. How does the triangle inequality help ensure a valid triangle exists?
4. Can you apply similar logic to find possible side lengths for any triangle?
5. What happens if the third side is a non-integer?

**Tip:** When working with triangle inequalities, always check all three conditions to ensure the sides form a valid triangle.

This problem explores how to use the triangle inequality theorem to find the smallest possible perimeter for a triangle with sides measuring 911 cm, 13 cm, and an unknown integer. Learn step-by-step how to apply the theorem and calculate the perimeter. Suitable for students in Grades 9-11.

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