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
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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.