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$In /_\DEF,DE=5 and EF=8. If the length of the third side is an integer, what is the greatest possible value for DF ? 5+8=12$

$2$ ) In $\triangle D E F, D E=5$ and $E F=8$ . If the length of the third side is an integer, what is the greatest possible value for $D F$ ? $\newline$ $5+8=12$

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Composition of linear functions: find a value

Full solution

Q.

2

) In

\triangle D E F, D E=5

and

E F=8

. If the length of the third side is an integer, what is the greatest possible value for

D F

?

\newline

5+8=12

Triangle Inequality Theorem: We know that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Applying Inequality to DE + EF: So, $DE + EF > DF$ , which means $5 + 8 > DF$ .
Calculating Upper Limit for DF: That gives us $13 > \text{DF}.$
Difference Inequality for $EF - DE$ : But we also know that the difference of the lengths of any two sides must be less than the length of the third side.
Calculating Lower Limit for DF: So, $EF - DE < DF$ , which means $8 - 5 < DF$ .
Determining Range for DF: That gives us $3 < \text{DF}.$
Maximum Integer Value for DF: Therefore, $DF$ must be greater than $3$ and less than $13$ .
Maximum Integer Value for DF: Therefore, $DF$ must be greater than $3$ and less than $13$ . Since $DF$ is an integer, the greatest possible integer value for $DF$ is $12$ .

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