Answer:
m∠DEF is 90°
Step-by-step explanation:
It is given that D(5, 7), E(4, 3), and F(8, 2) form the vertices of a triangle. Thus, using the distance formula, we have
[tex]DE=\sqrt{(3-7)^2+(4-5)^2}[/tex]
⇒[tex]DE=\sqrt{16+1}[/tex]
⇒[tex]DE=\sqrt{17}[/tex]
and [tex]EF=\sqrt{(2-3)^2+(8-4)^2}[/tex]
⇒[tex]EF=\sqrt{1+16}[/tex]
⇒[tex]EF=\sqrt{17}[/tex]
Also, [tex]DF=\sqrt{(2-7)^2+(8-6)^2}[/tex]
⇒[tex]DF=\sqrt{25+9}[/tex]
⇒[tex]DF=\sqrt{34}[/tex]
Now, [tex](DF)^2=(DE)^2+(EF)^2[/tex]
⇒[tex](\sqrt{34})^2=(\sqrt{17})^2+(\sqrt{17})^2[/tex]
⇒[tex]34=34[/tex]
Thus, Pythagoras theorem holds.
Hence, m∠DEF is 90°⇒ΔDEF is right angled triangle which is right angled at E.
