Chassily \( \triangle A B C \) by its sides. Then determine whether it is a right triangle. \( A(3,3), B(6,9), C(6,-3) \) The triangle can be classified by its sides as This triangle \( \square \) a right triangle.

To determine if triangle ABC is a right triangle, we can use the distance formula to find the lengths of its sides. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Calculating: - \(AB = \sqrt{(6 - 3)^2 + (9 - 3)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\) - \(BC = \sqrt{(6 - 6)^2 + (-3 - 9)^2} = \sqrt{0 + (-12)^2} = \sqrt{144} = 12\) - \(CA = \sqrt{(3 - 6)^2 + (3 - (-3))^2} = \sqrt{(-3)^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\) Now we have the side lengths: - \(AB = 3\sqrt{5}\) - \(BC = 12\) - \(CA = 3\sqrt{5}\) To check if it's a right triangle, we can use the Pythagorean theorem, where \(c^2 = a^2 + b^2\). Here \(c\) is the longest side \(BC=12\), and we need to check if \(12^2 = (3\sqrt{5})^2 + (3\sqrt{5})^2\): Calculating: - \(12^2 = 144\) - \((3\sqrt{5})^2 + (3\sqrt{5})^2 = 45 + 45 = 90\) Since \(144 \neq 90\), we conclude that this triangle is not a right triangle. The triangle can be classified by its sides as isosceles, since two sides are equal, \(AB\) and \(CA\). Thus, this triangle does not fit the definition of a right triangle!