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Answer :
To determine the range in which the length of the third side of a triangle must lie, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. With this in mind, let's apply this rule step-by-step.
Given two sides of the triangle are 29 units and 40 units, let's denote the length of the third side as [tex]\( x \)[/tex].
According to the triangle inequality theorem, we have:
1. The sum of the first and the second side must be greater than the third side:
[tex]\[
29 + 40 > x
\][/tex]
[tex]\[
x < 69
\][/tex]
2. The sum of the first and third side must be greater than the second side:
[tex]\[
29 + x > 40
\][/tex]
[tex]\[
x > 11
\][/tex]
3. The sum of the second and third side must be greater than the first side:
[tex]\[
40 + x > 29
\][/tex]
Since the value of [tex]\( x \)[/tex] that satisfies [tex]\( 40 + x > 29 \)[/tex] is already covered by [tex]\( x > 11 \)[/tex], we don't need a separate condition from step 3.
Combining these inequalities from step 1 and step 2, we get:
[tex]\[
11 < x < 69
\][/tex]
Therefore, the length of the third side must be greater than 11 and less than 69. This corresponds to option C: [tex]\( 11 < x < 69 \)[/tex].
Given two sides of the triangle are 29 units and 40 units, let's denote the length of the third side as [tex]\( x \)[/tex].
According to the triangle inequality theorem, we have:
1. The sum of the first and the second side must be greater than the third side:
[tex]\[
29 + 40 > x
\][/tex]
[tex]\[
x < 69
\][/tex]
2. The sum of the first and third side must be greater than the second side:
[tex]\[
29 + x > 40
\][/tex]
[tex]\[
x > 11
\][/tex]
3. The sum of the second and third side must be greater than the first side:
[tex]\[
40 + x > 29
\][/tex]
Since the value of [tex]\( x \)[/tex] that satisfies [tex]\( 40 + x > 29 \)[/tex] is already covered by [tex]\( x > 11 \)[/tex], we don't need a separate condition from step 3.
Combining these inequalities from step 1 and step 2, we get:
[tex]\[
11 < x < 69
\][/tex]
Therefore, the length of the third side must be greater than 11 and less than 69. This corresponds to option C: [tex]\( 11 < x < 69 \)[/tex].
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