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)

Question

Responses

4.04<x<80.1
4.04 less than x less than 80.1

4.67<x<42.3
4.67 less than x less than 42.3

4.4<x<80.73
4.4 less than x less than 80.73

4.67<x<80.73

Answer 1

To determine the range of values for the length of the third side of a triangle when the lengths of the other two sides are given, you can use the triangle inequality theorem. The triangle inequality states that for any triangle with sides a, b, and c, the following must hold true:

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

Let's denote the lengths of the two given sides as:

• Side 1 (a) = 42.7 mm
• Side 2 (b) = 38.03 mm
• Side 3 (x) = unknown side length

Applying the triangle inequalities, we have:

1. \( 42.7 + 38.03 > x \)

   • \( 80.73 > x \) or \( x < 80.73 \)

2. \( 42.7 + x > 38.03 \)

   • \( x > 38.03 - 42.7 \)
   • Since \( 38.03 - 42.7 = -4.67 \), this inequality gives no lower bound since \( x \) is always greater than a negative number.

3. \( 38.03 + x > 42.7 \)

   • \( x > 42.7 - 38.03 \)
   • \( x > 4.67 \)

From these inequalities, we can derive the useful bounds for x:

• From inequality (1): \( x < 80.73 \)
• From inequality (3): \( x > 4.67 \)

Putting these together, we find:

\[ 4.67 < x < 80.73 \]

Thus the correct response that matches this result is:

4.67 < x < 80.73