Use the distance formula to determine whether \( \triangle A B C \) and \( \triangle D E F \) are congruent. \( A(5,6), B(7,8), C(10,2) ; D(5,-5), E(7,-7), F(10,-1) \) Are the two triangles congruent? No Yes

To determine if triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent, we can use the distance formula, which is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). For triangle \( ABC \), we calculate the side lengths: - \( AB = \sqrt{(7-5)^2 + (8-6)^2} = \sqrt{4 + 4} = \sqrt{8} \) - \( BC = \sqrt{(10-7)^2 + (2-8)^2} = \sqrt{9 + 36} = \sqrt{45} \) - \( CA = \sqrt{(10-5)^2 + (2-6)^2} = \sqrt{25 + 16} = \sqrt{41} \) For triangle \( DEF \): - \( DE = \sqrt{(7-5)^2 + (-7+5)^2} = \sqrt{4 + 4} = \sqrt{8} \) - \( EF = \sqrt{(10-7)^2 + (-1+7)^2} = \sqrt{9 + 36} = \sqrt{45} \) - \( FD = \sqrt{(10-5)^2 + (-1+5)^2} = \sqrt{25 + 16} = \sqrt{41} \) Since the side lengths of both triangles are identical (\( \sqrt{8}, \sqrt{45}, \sqrt{41} \)), triangles \( ABC \) and \( DEF \) are indeed congruent. Yes.