
If the perimeter of the triangle is 57, what is the length of the *shortest* side of the triangle?

Responses

7

7

14

14

21

21


To determine the shortest side of a triangle given its perimeter, we must make an assumption that the sides of the triangle are positive integers. Since the perimeter is 57, the sum of the three sides should equal 57. 

Let's denote the sides of the triangle as \(a \leq b \leq c\), where \(a\) is the shortest side. The possible lengths for the shortest side \(a\) should satisfy the triangle inequality rules:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Given the response choices are: \(7, 7, 14, 14, 21, 21\), let's examine each one to see which could reasonably be the shortest side.

### Reasoning Through the Options:

1. **7 as the shortest side:**
   - If \(a = 7\), we have:
     \[
     b + c = 57 - 7 = 50
     \]
     This means \(b\) and \(c\) must be two numbers that sum up to 50 and are both greater than or equal to 7.

2. **14 as the shortest side:**
   - If \(a = 14\), we have:
     \[
     b + c = 57 - 14 = 43
     \]
     This means \(b\) and \(c\) must be two numbers that sum up to 43 and are both greater than or equal to 14.

3. **21 as the shortest side:**
   - If \(a = 21\), we have:
     \[
     b + c = 57 - 21 = 36
     \]
     This means \(b\) and \(c\) must be two numbers that sum up to 36 and are both greater than or equal to 21.

### Conclusion:

The smallest possible value for \(a\) (shortest side) while satisfying the triangle inequality rules is \(7\). 

Thus, the length of the shortest side of the triangle is **7**.

Would you like more details or have any questions? Here are some related questions:

1. What are the possible lengths for the other two sides of the triangle?
2. How do the triangle inequality rules apply in different scenarios?
3. What would happen if the perimeter were different (e.g., 60)?
4. How does the classification of triangles (equilateral, isosceles, scalene) change with different side lengths?
5. Can you have a triangle with any three arbitrary side lengths?

**Tip:** Always check if the side lengths of a triangle satisfy the triangle inequality rules to determine if a valid triangle can be formed.

If the perimeter of the triangle is 57, what is the length of the shortest side of the triangle?

Learn how to determine the shortest side of a triangle with a perimeter of 57 using the Triangle Inequality Theorem. This problem covers key geometric concepts and calculations suitable for students in Grades 7-9.

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