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Answer :
To solve the problem, let's break it down step-by-step:
1. Understand the problem: We have a triangle with a perimeter of 32 inches. We need to find the length of the shortest side given the following:
- The shortest side (let's call it [tex]\(x\)[/tex]) is [tex]\( \frac{1}{3} \)[/tex] of the longest side.
- The shortest side is also 7 inches less than the remaining side.
2. Define the sides based on the information:
- Shortest side: [tex]\(x\)[/tex]
- Longest side: Since the shortest side is [tex]\(\frac{1}{3}\)[/tex] of the longest side, the longest side is [tex]\(3x\)[/tex].
- Remaining side: We know that the shortest side is 7 inches less than this side, so the remaining side is [tex]\(x + 7\)[/tex].
3. Set up the equation for the perimeter: The perimeter is the sum of all the sides. According to the problem, the perimeter is 32 inches, so:
[tex]\[
x + 3x + (x + 7) = 32
\][/tex]
4. Simplify the equation:
- Combine like terms: [tex]\(x + 3x + x + 7 = 5x + 7\)[/tex]
- Set the equation:
[tex]\[
5x + 7 = 32
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Subtract 7 from both sides to isolate the terms with [tex]\(x\)[/tex]:
[tex]\[
5x = 25
\][/tex]
- Divide by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 5
\][/tex]
6. Conclusion: The length of the shortest side is 5 inches.
This solution explains how we derive the result using logical steps based on the given conditions.
1. Understand the problem: We have a triangle with a perimeter of 32 inches. We need to find the length of the shortest side given the following:
- The shortest side (let's call it [tex]\(x\)[/tex]) is [tex]\( \frac{1}{3} \)[/tex] of the longest side.
- The shortest side is also 7 inches less than the remaining side.
2. Define the sides based on the information:
- Shortest side: [tex]\(x\)[/tex]
- Longest side: Since the shortest side is [tex]\(\frac{1}{3}\)[/tex] of the longest side, the longest side is [tex]\(3x\)[/tex].
- Remaining side: We know that the shortest side is 7 inches less than this side, so the remaining side is [tex]\(x + 7\)[/tex].
3. Set up the equation for the perimeter: The perimeter is the sum of all the sides. According to the problem, the perimeter is 32 inches, so:
[tex]\[
x + 3x + (x + 7) = 32
\][/tex]
4. Simplify the equation:
- Combine like terms: [tex]\(x + 3x + x + 7 = 5x + 7\)[/tex]
- Set the equation:
[tex]\[
5x + 7 = 32
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Subtract 7 from both sides to isolate the terms with [tex]\(x\)[/tex]:
[tex]\[
5x = 25
\][/tex]
- Divide by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 5
\][/tex]
6. Conclusion: The length of the shortest side is 5 inches.
This solution explains how we derive the result using logical steps based on the given conditions.
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