The perimeter of a scalene triangle is [tex]$14.5 \, \text{cm}[tex]$[/tex]. The longest side is twice that of the shortest side. Which equation can be used to find the side lengths if the longest side measures [tex]$[/tex]6.2 \, \text{cm}$[/tex]?

Choose the correct answer:

A. [tex]6.2 + b = 14.5[/tex]

B. [tex]9.3 + b = 14.5[/tex]

C. [tex]12.4 + b = 14.5[/tex]

D. [tex]18.6 + b = 14.5[/tex]



Answer :

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]