College

We appreciate your visit to Select the correct answer A triangle has one side of length 29 units and another of length 40 units Determine the range in which the. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Select the correct answer.

A triangle has one side of length 29 units and another of length 40 units. Determine the range in which the length of the third side must lie.

A. [tex]-11 \ < \ x \ < \ 69[/tex]

B. [tex]11 \leq x \leq 69[/tex]

C. [tex]11 \ < \ x \ < \ 69[/tex]

D. [tex]-11 \leq x \leq 69[/tex]

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].

Thanks for taking the time to read Select the correct answer A triangle has one side of length 29 units and another of length 40 units Determine the range in which the. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada