Question

is \\( \triangle def \\) an equilateral triangle? justify your answer.
yes, all sides are congruent.
yes, \\( \overline { e f } \\) is congruent to \\( \overline { d f } \\).
no, \\( \overline { e f } \\) is not congruent to \\( \overline { d f } \\).
no, \\( \overline { d e } \\) is not congruent to \\( \overline { e f } \\).

Explanation:

Step1: Calculate the length of \(DE\)

The coordinates of \(D(3,1)\) and \(E(7,1)\). Using the distance formula for horizontal line \(d=\vert x_2 - x_1\vert\) (since \(y_1=y_2 = 1\)), \(DE=\vert7 - 3\vert=4\).

Step2: Calculate the length of \(EF\)

The coordinates of \(E(7,1)\) and \(F(5, - 2)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(EF=\sqrt{(5 - 7)^2+(-2 - 1)^2}=\sqrt{(-2)^2+(-3)^2}=\sqrt{4 + 9}=\sqrt{13}\approx3.61\).

Step3: Compare the lengths

Since \(DE = 4\) and \(EF\approx3.61\), \(DE
eq EF\).

Answer:

No, \(\overline{DE}\) is not congruent to \(\overline{EF}\).