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Answer :
To determine the range in which the length of the third side of the triangle must lie, we can apply the Triangle Inequality Theorem. This theorem states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Given the sides [tex]\( a = 29 \)[/tex] and [tex]\( b = 40 \)[/tex], let [tex]\( x \)[/tex] be the third side.
Let's determine the possible range for [tex]\( x \)[/tex]:
1. Sum of two sides greater than the third:
- [tex]\( 29 + 40 > x \)[/tex]
Simplifies to: [tex]\( x < 69 \)[/tex]
2. Each side must be less than the sum of the other two sides:
- [tex]\( 29 + x > 40 \)[/tex]
Simplifies to: [tex]\( x > 40 - 29 \)[/tex]
Thus: [tex]\( x > 11 \)[/tex]
- [tex]\( 40 + x > 29 \)[/tex]
This inequality is always true if [tex]\( x > 0 \)[/tex] because [tex]\( 40 > 29 \)[/tex].
Combining these inequalities, the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
So, the correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Given the sides [tex]\( a = 29 \)[/tex] and [tex]\( b = 40 \)[/tex], let [tex]\( x \)[/tex] be the third side.
Let's determine the possible range for [tex]\( x \)[/tex]:
1. Sum of two sides greater than the third:
- [tex]\( 29 + 40 > x \)[/tex]
Simplifies to: [tex]\( x < 69 \)[/tex]
2. Each side must be less than the sum of the other two sides:
- [tex]\( 29 + x > 40 \)[/tex]
Simplifies to: [tex]\( x > 40 - 29 \)[/tex]
Thus: [tex]\( x > 11 \)[/tex]
- [tex]\( 40 + x > 29 \)[/tex]
This inequality is always true if [tex]\( x > 0 \)[/tex] because [tex]\( 40 > 29 \)[/tex].
Combining these inequalities, the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[ 11 < x < 69 \][/tex]
So, the correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]
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