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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. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's denote the sides of the triangle as [tex]\( a = 29 \)[/tex], [tex]\( b = 40 \)[/tex], and [tex]\( c \)[/tex] as the unknown third side. The triangle inequality theorem gives us three conditions:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Substituting the known values, these inequalities become:
1. [tex]\( 29 + 40 > c \)[/tex]
[tex]\[ c < 69 \][/tex]
2. [tex]\( 29 + c > 40 \)[/tex]
[tex]\[ c > 11 \][/tex]
3. [tex]\( 40 + c > 29 \)[/tex]
[tex]\[ c > -11 \][/tex]
(Although this condition is always true when [tex]\( c > 11 \)[/tex], since sides cannot be negative.)
Taking the intersection of these conditions, the third side [tex]\( c \)[/tex] must satisfy:
[tex]\[ 11 < c < 69 \][/tex]
Therefore, the correct range for the length of the third side is:
C. [tex]\(11
Let's denote the sides of the triangle as [tex]\( a = 29 \)[/tex], [tex]\( b = 40 \)[/tex], and [tex]\( c \)[/tex] as the unknown third side. The triangle inequality theorem gives us three conditions:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Substituting the known values, these inequalities become:
1. [tex]\( 29 + 40 > c \)[/tex]
[tex]\[ c < 69 \][/tex]
2. [tex]\( 29 + c > 40 \)[/tex]
[tex]\[ c > 11 \][/tex]
3. [tex]\( 40 + c > 29 \)[/tex]
[tex]\[ c > -11 \][/tex]
(Although this condition is always true when [tex]\( c > 11 \)[/tex], since sides cannot be negative.)
Taking the intersection of these conditions, the third side [tex]\( c \)[/tex] must satisfy:
[tex]\[ 11 < c < 69 \][/tex]
Therefore, the correct range for the length of the third side is:
C. [tex]\(11
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