\( A(2,3), B(5,4), C(6,7) \) the \( \triangle A B C \) is

Below is the step‐by‐step solution to classify the triangle with vertices \( A(2,3) \), \( B(5,4) \), and \( C(6,7) \). --- ### 1. Compute the side lengths using the distance formula The distance formula between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is given by: \[ d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \] #### Side \( AB \): \[ AB = \sqrt{(5-2)^2 + (4-3)^2} = \sqrt{3^2+1^2} = \sqrt{9+1} = \sqrt{10} \] #### Side \( BC \): \[ BC = \sqrt{(6-5)^2 + (7-4)^2} = \sqrt{1^2+3^2} = \sqrt{1+9} = \sqrt{10} \] #### Side \( AC \): \[ AC = \sqrt{(6-2)^2 + (7-3)^2} = \sqrt{4^2+4^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2} \] --- ### 2. Analyze the side lengths From the calculations: - \( AB = \sqrt{10} \) - \( BC = \sqrt{10} \) - \( AC = 4\sqrt{2} \) Since \( AB = BC \) and \( AC \) is different, the triangle has **two equal sides**. --- ### 3. Conclusion Because the triangle has two sides of equal length, \( \triangle ABC \) is an **isosceles triangle**. --- ### Final Answer \( \triangle ABC \) is an **isosceles triangle**.