Explanation:
Part A.
To get from D to E, we go 5 units right and 2 units up, then we can write the following equation to find the length DE
[tex]5^2+2^2=DE^2[/tex]Solving for DE, we get
[tex]\begin{gathered} DE^2=5^2+2^2 \\ DE^2=25+4 \\ DE^2=29 \\ DE=\sqrt{29} \\ DE=5.4 \end{gathered}[/tex]Therefore, DE = 5.4 units
Part B.
To get from D to F, we go 5 units right and 4 units down, so we can write the following equation to find the length of DF
[tex]5^2+4^2=DF^2[/tex]Solving for DF, we get
[tex]\begin{gathered} DF^2=5^2+4^2 \\ DF^2=25+16 \\ DF^2=41 \\ DF=\sqrt{41} \\ DF=6.4 \end{gathered}[/tex]Therefore, the length of DF is 6.4 units
Part C.
Then, the perimeter is the sum of all the sides of the triangle, so
Perimeter = DE + DF + FE
Perimeter = 5.4 + 6.4 + 6
Perimeter = 17.8 units