Question
Sides of a Triangle Quick Check 5 of 55 of 5 Items Question Find the range of values for the third side of a triangle if two of its sides measure 42.7 mm and 38.03 mm.(1 point) Responses 4.67<x<42.3 4.67 less than x less than 42.3 4.67<x<80.73 4.67 less than x less than 80.73 4.4<x<80.73 4.4 less than x less than 80.73 4.04<x<80.1 4.04 less than x less than 80.1 Skip to navigation page 15 of 15
Answers
GPT-4o mini
To find the range of values for the third side of a triangle when two sides are given, we can use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\):
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Here, we need to find the possible values for the third side \(c\) given that the other two sides are \(a = 42.7 \, \text{mm}\) and \(b = 38.03 \, \text{mm}\).
### Using the triangle inequality:
1. \(42.7 + 38.03 > c\)
- \(80.73 > c\)
- Hence, \(c < 80.73\)
2. \(42.7 + c > 38.03\)
- This simplifies to \(c > 38.03 - 42.7\)
- \(c > -4.67\) (not really restrictive in positive lengths)
3. \(38.03 + c > 42.7\)
- This simplifies to \(c > 42.7 - 38.03\)
- \(c > 4.67\)
### Conclusion:
Combining these inequalities, we have:
\[ 4.67 < c < 80.73 \]
Therefore, the range of values for the third side of the triangle is:
\[ \text{4.67 < c < 80.73} \]
The answer choice that corresponds to this result is:
**4.67 < x < 80.73**.
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Here, we need to find the possible values for the third side \(c\) given that the other two sides are \(a = 42.7 \, \text{mm}\) and \(b = 38.03 \, \text{mm}\).
### Using the triangle inequality:
1. \(42.7 + 38.03 > c\)
- \(80.73 > c\)
- Hence, \(c < 80.73\)
2. \(42.7 + c > 38.03\)
- This simplifies to \(c > 38.03 - 42.7\)
- \(c > -4.67\) (not really restrictive in positive lengths)
3. \(38.03 + c > 42.7\)
- This simplifies to \(c > 42.7 - 38.03\)
- \(c > 4.67\)
### Conclusion:
Combining these inequalities, we have:
\[ 4.67 < c < 80.73 \]
Therefore, the range of values for the third side of the triangle is:
\[ \text{4.67 < c < 80.73} \]
The answer choice that corresponds to this result is:
**4.67 < x < 80.73**.