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

Answer 1

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.