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To determine which equation can be used to find the value of [tex]\( b \)[/tex] in a scalene triangle where one side [tex]\( a \)[/tex] measures 8.7 cm and the perimeter is 54.6 cm, you need to examine each option. Here's a detailed step-by-step analysis:
### Step-by-Step Analysis:
1. Understanding the Perimeter:
- The perimeter of a triangle is the sum of all its side lengths.
- Given perimeter [tex]\( P = 54.6 \)[/tex] cm.
- Given one side [tex]\( a = 8.7 \)[/tex] cm.
2. Setting up the Equations:
- The equation to find [tex]\( b \)[/tex] should involve the perimeter and the given side [tex]\( a \)[/tex].
3. Evaluating Each Option:
- Option 1: [tex]\( 8.7 + b = 54.6 \)[/tex]
- This equation implies that the remaining perimeter after excluding side [tex]\( a \)[/tex] is the side [tex]\( b \)[/tex].
- Solve for [tex]\( b \)[/tex]:
[tex]\[ b = 54.6 - 8.7 = 45.9 \][/tex]
- This seems logical since only one side [tex]\( a \)[/tex] is given and [tex]\( b \)[/tex] needs to be found.
- Option 2: [tex]\( 17.4 + b = 54.6 \)[/tex]
- This equation implies [tex]\( 17.4 \)[/tex] (double of [tex]\( a \)[/tex]) and side [tex]\( b \)[/tex] sum to the perimeter.
- However, since 17.4 is not a given side but rather [tex]\( 2a \)[/tex], solving for [tex]\( b \)[/tex] yields:
[tex]\[ b = 54.6 - 17.4 = 37.2 \][/tex]
- This could make sense in context because it involves multiple sides of the triangle equating to the perimeter.
- Option 3: [tex]\( 26.1 + b = 54.6 \)[/tex]
- This equation uses [tex]\( 26.1 \)[/tex], which isn't related to the given side length or any logical combination of side [tex]\( a \)[/tex].
- Solving for [tex]\( b \)[/tex]:
[tex]\[ b = 54.6 - 26.1 = 28.5 \][/tex]
- This is an arbitrary value not connected to our known side length [tex]\( a \)[/tex].
- Option 4: [tex]\( 248 + h = 546 \)[/tex]
- This equation is not compatible with the given metric system or the side lengths concerned in the problem.
- The equation is irrelevant to the triangle with side [tex]\( a \)[/tex] and given perimeter.
### Conclusion:
After analyzing all options, Option 2: [tex]\( 17.4 + b = 54.6 \)[/tex] can be used to find the value of [tex]\( b \)[/tex] because it logically considers a new side [tex]\( b \)[/tex] and a legitimate perimeter derived from the given values. This equation fits the conditions of the problem involving triangle perimeter calculation with given side lengths.
To summarize, the equation that best fits and can be used to find the value of [tex]\( b \)[/tex] is:
[tex]\[ \boxed{17.4 + b = 54.6} \][/tex]
### Step-by-Step Analysis:
1. Understanding the Perimeter:
- The perimeter of a triangle is the sum of all its side lengths.
- Given perimeter [tex]\( P = 54.6 \)[/tex] cm.
- Given one side [tex]\( a = 8.7 \)[/tex] cm.
2. Setting up the Equations:
- The equation to find [tex]\( b \)[/tex] should involve the perimeter and the given side [tex]\( a \)[/tex].
3. Evaluating Each Option:
- Option 1: [tex]\( 8.7 + b = 54.6 \)[/tex]
- This equation implies that the remaining perimeter after excluding side [tex]\( a \)[/tex] is the side [tex]\( b \)[/tex].
- Solve for [tex]\( b \)[/tex]:
[tex]\[ b = 54.6 - 8.7 = 45.9 \][/tex]
- This seems logical since only one side [tex]\( a \)[/tex] is given and [tex]\( b \)[/tex] needs to be found.
- Option 2: [tex]\( 17.4 + b = 54.6 \)[/tex]
- This equation implies [tex]\( 17.4 \)[/tex] (double of [tex]\( a \)[/tex]) and side [tex]\( b \)[/tex] sum to the perimeter.
- However, since 17.4 is not a given side but rather [tex]\( 2a \)[/tex], solving for [tex]\( b \)[/tex] yields:
[tex]\[ b = 54.6 - 17.4 = 37.2 \][/tex]
- This could make sense in context because it involves multiple sides of the triangle equating to the perimeter.
- Option 3: [tex]\( 26.1 + b = 54.6 \)[/tex]
- This equation uses [tex]\( 26.1 \)[/tex], which isn't related to the given side length or any logical combination of side [tex]\( a \)[/tex].
- Solving for [tex]\( b \)[/tex]:
[tex]\[ b = 54.6 - 26.1 = 28.5 \][/tex]
- This is an arbitrary value not connected to our known side length [tex]\( a \)[/tex].
- Option 4: [tex]\( 248 + h = 546 \)[/tex]
- This equation is not compatible with the given metric system or the side lengths concerned in the problem.
- The equation is irrelevant to the triangle with side [tex]\( a \)[/tex] and given perimeter.
### Conclusion:
After analyzing all options, Option 2: [tex]\( 17.4 + b = 54.6 \)[/tex] can be used to find the value of [tex]\( b \)[/tex] because it logically considers a new side [tex]\( b \)[/tex] and a legitimate perimeter derived from the given values. This equation fits the conditions of the problem involving triangle perimeter calculation with given side lengths.
To summarize, the equation that best fits and can be used to find the value of [tex]\( b \)[/tex] is:
[tex]\[ \boxed{17.4 + b = 54.6} \][/tex]
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