Answer :
To determine the range in which the length of the third side of the triangle must lie, we'll use the Triangle Inequality Theorem. This theorem states:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given the triangle sides with lengths 29 units and 40 units, let's denote the length of the third side as [tex]\( x \)[/tex]. We apply the triangle inequality for all combinations:
1. The first inequality comes from ensuring that the sum of the two given sides is greater than [tex]\( x \)[/tex]:
[tex]\[
29 + 40 > x
\][/tex]
[tex]\[
69 > x \quad \text{or} \quad x < 69
\][/tex]
2. The second inequality comes from ensuring that the sum of one given side and [tex]\( x \)[/tex] is greater than the other side:
[tex]\[
29 + x > 40
\][/tex]
[tex]\[
x > 40 - 29
\][/tex]
[tex]\[
x > 11
\][/tex]
3. The third inequality comes from ensuring that the sum of the other given side and [tex]\( x \)[/tex] is greater than the first side:
[tex]\[
40 + x > 29
\][/tex]
However, this simplifies to the same inverse condition:
[tex]\[
x > 29 - 40
\][/tex]
which confirms [tex]\( x > -11 \)[/tex], but since [tex]\( x > 11 \)[/tex] is already stronger, we use [tex]\( x > 11 \)[/tex].
Combining these results, we find that the length of the third side must satisfy:
[tex]\[
11 < x < 69
\][/tex]
Therefore, the correct choice for the range in which the third side of the triangle must lie is option C: [tex]\( 11 < x < 69 \)[/tex].
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given the triangle sides with lengths 29 units and 40 units, let's denote the length of the third side as [tex]\( x \)[/tex]. We apply the triangle inequality for all combinations:
1. The first inequality comes from ensuring that the sum of the two given sides is greater than [tex]\( x \)[/tex]:
[tex]\[
29 + 40 > x
\][/tex]
[tex]\[
69 > x \quad \text{or} \quad x < 69
\][/tex]
2. The second inequality comes from ensuring that the sum of one given side and [tex]\( x \)[/tex] is greater than the other side:
[tex]\[
29 + x > 40
\][/tex]
[tex]\[
x > 40 - 29
\][/tex]
[tex]\[
x > 11
\][/tex]
3. The third inequality comes from ensuring that the sum of the other given side and [tex]\( x \)[/tex] is greater than the first side:
[tex]\[
40 + x > 29
\][/tex]
However, this simplifies to the same inverse condition:
[tex]\[
x > 29 - 40
\][/tex]
which confirms [tex]\( x > -11 \)[/tex], but since [tex]\( x > 11 \)[/tex] is already stronger, we use [tex]\( x > 11 \)[/tex].
Combining these results, we find that the length of the third side must satisfy:
[tex]\[
11 < x < 69
\][/tex]
Therefore, the correct choice for the range in which the third side of the triangle must lie is option C: [tex]\( 11 < x < 69 \)[/tex].
