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Answer :
To find the range of values for the third side of a triangle when given two sides, we can use the Triangle Inequality Theorem. Let's walk through the steps to find the possible values for the third side, [tex]\( x \)[/tex], when the two sides are 184 and 208.
The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the third side. This gives us three inequalities:
1. [tex]\( 184 + 208 > x \)[/tex]
2. [tex]\( 184 + x > 208 \)[/tex]
3. [tex]\( 208 + x > 184 \)[/tex]
Now, let's solve each inequality:
1. From [tex]$184 + 208 > x$[/tex], we get:
[tex]\[
392 > x
\][/tex]
This means [tex]\( x < 392 \)[/tex].
2. From [tex]$184 + x > 208$[/tex], we subtract 184 from both sides:
[tex]\[
x > 24
\][/tex]
3. From [tex]$208 + x > 184$[/tex], we subtract 208 from both sides, but notice that this inequality is always true since 24 (which is our minimum value from step 2) is greater than -24:
[tex]\[
x > -24
\][/tex]
Considering the valid and overlapping constraints, the range of possible values for the third side [tex]\( x \)[/tex] is:
[tex]\[ 24 < x < 392 \][/tex]
This means the third side must be greater than 24 but less than 392 for the sides to form a valid triangle.
The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the third side. This gives us three inequalities:
1. [tex]\( 184 + 208 > x \)[/tex]
2. [tex]\( 184 + x > 208 \)[/tex]
3. [tex]\( 208 + x > 184 \)[/tex]
Now, let's solve each inequality:
1. From [tex]$184 + 208 > x$[/tex], we get:
[tex]\[
392 > x
\][/tex]
This means [tex]\( x < 392 \)[/tex].
2. From [tex]$184 + x > 208$[/tex], we subtract 184 from both sides:
[tex]\[
x > 24
\][/tex]
3. From [tex]$208 + x > 184$[/tex], we subtract 208 from both sides, but notice that this inequality is always true since 24 (which is our minimum value from step 2) is greater than -24:
[tex]\[
x > -24
\][/tex]
Considering the valid and overlapping constraints, the range of possible values for the third side [tex]\( x \)[/tex] is:
[tex]\[ 24 < x < 392 \][/tex]
This means the third side must be greater than 24 but less than 392 for the sides to form a valid triangle.
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