High School

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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 possible length of the third side in a triangle with two sides of length [tex]$29$[/tex] and [tex]$40$[/tex], we use the triangle inequality theorem. This theorem states that the length of any side must be less than the sum and greater than the difference of the other two sides. For the third side, [tex]$x$[/tex], this gives:

[tex]$$
|29 - 40| < x < 29 + 40.
$$[/tex]

Calculating the bounds, we have:

1. The lower bound:
[tex]$$
|29 - 40| = 11,
$$[/tex]
so [tex]$x > 11$[/tex].

2. The upper bound:
[tex]$$
29 + 40 = 69,
$$[/tex]
so [tex]$x < 69$[/tex].

Therefore, the length of the third side must satisfy:

[tex]$$
11 < x < 69.
$$[/tex]

This corresponds to option C.

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