Triangle \( A B C \) has the following vertices: - \( A(7,-2) \) - \( B(-1,-8) \) - \( C(5,0) \) Is triangle \( A B C \) an isosceles triangle, and why? Choose 1 answer: (A) Yes, because \( A B=B C \). (B) Yes, because \( A C=B C \). (C) No, because \( \overline{A C} \) is longer than \( \overline{A B} \). (D) No, because \( \triangle A B C \) doesn't have a pair of sides with equal lengths.

To determine whether triangle \( ABC \) is isosceles, we first need to calculate the lengths of its sides: 1. Length of \( AB \): \[ AB = \sqrt{(7 - (-1))^2 + (-2 - (-8))^2} = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \] 2. Length of \( BC \): \[ BC = \sqrt{(-1 - 5)^2 + (-8 - 0)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \] 3. Length of \( AC \): \[ AC = \sqrt{(7 - 5)^2 + (-2 - 0)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 \] Now, comparing the lengths: - \( AB = 10 \) - \( BC = 10 \) - \( AC \approx 2.83 \) Since \( AB \) and \( BC \) are equal, triangle \( ABC \) is indeed isosceles. Therefore, the answer is **(A) Yes, because \( AB = BC \)**. Now, let's explore some more interesting aspects related to this triangle: Calculating the lengths of sides reveals not just geometric properties but also a cool insight. An isosceles triangle has two equal sides, and in this case, those sides happen to be longer than the remaining side. Think of it as a balanced structure; they hold each other up! Triangles like this one are commonly used in architecture and engineering to ensure stability. They’re the reason we can have wide spans in bridges and majestic rooftop designs. Understanding their properties not only solves geometric puzzles but also helps inventors and builders create lasting structures!