To solve the problem, let's understand the relationship between the given side lengths and the perimeter of the triangle.
1. Identify the given values:
- The longest side of the triangle: \(6.2 \, \text{cm}\)
- The perimeter of the triangle: \(14.5 \, \text{cm}\)
- The longest side is twice the shortest side.
2. Determine the algebraic relationship:
Let's denote:
- The shortest side as \( a \)
- The middle side as \( b \)
From the problem, we know that the longest side is twice the shortest side:
[tex]\[
\text{Longest side} = 2a = 6.2 \, \text{cm}
\][/tex]
Solving for \( a \), we get:
[tex]\[
a = \frac{6.2}{2} = 3.1 \, \text{cm}
\][/tex]
3. Express the perimeter equation:
The perimeter of the triangle is the sum of all its sides:
[tex]\[
\text{Perimeter} = \text{Shortest side} + \text{Middle side} + \text{Longest side}
\][/tex]
Substituting the known values, we get:
[tex]\[
14.5 = a + b + 6.2
\][/tex]
Since \( a \) is \( 3.1 \, \text{cm} \):
[tex]\[
14.5 = 3.1 + b + 6.2
\][/tex]
4. Find \( b \):
Rearrange the equation to solve for \( b \):
[tex]\[
b = 14.5 - 6.2
\][/tex]
[tex]\[
b = 14.5 - 6.2 = 8.3
\][/tex]
This step confirms that the calculations are correctly done and the middle side \( b \) has been identified as \( 8.3 \, \text{cm} \).
5. Verify the correct equation:
Let's check the given multiple-choice options to see which equation is consistent with our determined perimeter value.
- \( 6.2 + b = 14.5 \)
- \( 9.3 + b = 14.5 \)
- \( 12.4 + b = 14.5 \)
- \( 18.6 + b = 14.5 \)
Substituting \( b = 8.3 \) into each equation:
- \( 6.2 + 8.3 = 14.5 \), which is correct.
- \( 9.3 + 8.3 = 17.6 \), which is incorrect.
- \( 12.4 + 8.3 = 20.7 \), which is incorrect.
- \( 18.6 + 8.3 = 26.9 \), which is incorrect.
Therefore, the correct equation which can be used to find the side lengths, given the conditions of the problem, is:
[tex]\[
\boxed{6.2 + b = 14.5}
\][/tex]
