6. Determine whether \( \triangle D E F \) with vertices \( D(6,-6), E(39,-12) \), and \( F(24,18) \) is scalene, isosceles, or equilateral. For Exercises \( 7-10 \), graph each quadrilateral using the given vertices. Then use the

To determine the type of triangle \( \triangle D E F \), we first need to calculate the lengths of its sides using the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Calculating the lengths of each side: - For side \( DE \): \[ DE = \sqrt{(39 - 6)^2 + (-12 + 6)^2} = \sqrt{33^2 + (-6)^2} = \sqrt{1089 + 36} = \sqrt{1125} \approx 33.54 \] - For side \( EF \): \[ EF = \sqrt{(39 - 24)^2 + (-12 - 18)^2} = \sqrt{15^2 + (-30)^2} = \sqrt{225 + 900} = \sqrt{1125} \approx 33.54 \] - For side \( FD \): \[ FD = \sqrt{(24 - 6)^2 + (18 + 6)^2} = \sqrt{18^2 + 24^2} = \sqrt{324 + 576} = \sqrt{900} = 30 \] Now, we compare the lengths: - \( DE \approx 33.54 \) - \( EF \approx 33.54 \) - \( FD = 30 \) Since \( DE \) and \( EF \) are equal, this makes \( \triangle DEF \) an isosceles triangle.