Triangle DEF, with vertices D(3,-7), E(6,-6), and F(2,-3), is drawn inside a rectangle
Let's calculate the length of the sides of the rectangle using the coordinates of the triangle vertices.
Length of side DE:
DE = sqrt((6-3)^2 + (-6-(-7))^2)
DE = sqrt(3^2 + 1^2)
DE = sqrt(9 + 1)
DE = sqrt(10)
Length of side EF:
EF = sqrt((2-6)^2 + (-3-(-6))^2)
EF = sqrt((-4)^2 + 3^2)
EF = sqrt(16 + 9)
EF = sqrt(25)
EF = 5
Since DE = EF, the triangle DEF is an isosceles triangle.
Now, let's calculate the length of side DF:
DF = sqrt((2-3)^2 + (-3-(-7))^2)
DF = sqrt((-1)^2 + 4^2)
DF = sqrt(1 + 16)
DF = sqrt(17)
Since DE ≠ DF, the triangle DEF is not an equilateral triangle.
Therefore, we can conclude that the triangle DEF is an isoscejson triangle with sides of length DE = EF = sqrt(10) and DF = sqrt(17). The triangle is drawn inside a rectangle that surrounds it.