1. **Problem Statement:** Determine the type of triangle \(\triangle ABC\) formed by points \(A(-4,2)\), \(B(-4,-6)\), and \(C(2,-6)\). Also, check if it is a right triangle. 2. **Step 1: Calculate the lengths of sides \(AB\), \(BC\), and \(AC\).** Use the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - \(AB = \sqrt{(-4 - (-4))^2 + (-6 - 2)^2} = \sqrt{0 + (-8)^2} = \sqrt{64} = 8\) - \(BC = \sqrt{(2 - (-4))^2 + (-6 - (-6))^2} = \sqrt{6^2 + 0} = 6\) - \(AC = \sqrt{(2 - (-4))^2 + (-6 - 2)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\) 3. **Step 2: Classify the triangle by side lengths.** - All sides are different: 8, 6, and 10. - Therefore, \(\triangle ABC\) is **scalene**. 4. **Step 3: Check if \(\triangle ABC\) is a right triangle using the Pythagorean theorem.** - Check if \(AB^2 + BC^2 = AC^2\): $$8^2 + 6^2 = 64 + 36 = 100 = 10^2$$ - Since this holds true, \(\triangle ABC\) is a **right triangle** with the right angle at \(B\). Final answer: \(\triangle ABC\) is a **scalene right triangle**.