I BapnaHT 1, HağnHTe KOopIMHaTEI BeKTODa eciH A(-7; 6), B(-1: 2). 2. HaňIHTe IHHY BeKIopa AB ecIH A(-7; 6), B(-1;2). 3. HažAHTe KOopHaT LI TOKIt , KOTopas ABIAETCA CepeIH 40 OTDe3Ka N ecIH M(6; -5), N(3; -9) 4. HaňnHTe paccTOIHHE MeKTV TOHKaMH M H N, T.e. UHHY oTpe3ka MN, ec M(6; -5), N(3; -9) 5. HaňIHTe MeKaHy CD AB ABC, BepiHHbI TpeyrOHbHMKa KOTOporo HMCK1 KOopIHHaTbI: A(-1; 2), B(5: -6), C(6: 4)

Let's solve each of the problems step by step based on the provided data points. ### 1. Find the Length of Segment AB Points: A(-7, 6), B(-1, 2) To find the length of a segment between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), you can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the values: \[ d = \sqrt{((-1) - (-7))^2 + (2 - 6)^2} = \sqrt{(6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \] ### 2. Find the Midpoint of Segment AB Points: A(-7, 6), B(-1, 2) The midpoint \( M \) of a segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the values: \[ M = \left( \frac{-7 + (-1)}{2}, \frac{6 + 2}{2} \right) = \left( \frac{-8}{2}, \frac{8}{2} \right) = (-4, 4) \] ### 3. Find the Slope of Line Segment MN Points: M(6, -5), N(3, -9) The slope \( m \) of a line through points \( M(x_1, y_1) \) and \( N(x_2, y_2) \) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the values: \[ m = \frac{-9 - (-5)}{3 - 6} = \frac{-4}{-3} = \frac{4}{3} \] ### 4. Find the Length of Segment MN Points: M(6, -5), N(3, -9) Using the distance formula: \[ d = \sqrt{(3 - 6)^2 + (-9 - (-5))^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] ### 5. Calculate the Area of Triangle ABC Points: A(-1, 2), B(5, -6), C(6, 4) The area \( A \) of a triangle given vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), \( C(x_3, y_3) \) can be found using the formula: \[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Substituting the values: \[ A = \frac{1}{2} \left| -1(-6 - 4) + 5(4 - 2) + 6(2 - (-6)) \right| \] Calculating it step-by-step: \[ = \frac{1}{2} \left| -1(-10) + 5(2) + 6(8) \right| = \frac{1}{2} \left| 10 + 10 + 48 \right| = \frac{1}{2} \left| 68 \right| = 34 \] ### Summary of the Results 1. Length of AB: \( 2\sqrt{13} \) 2. Midpoint of AB: \( (-4, 4) \) 3. Slope of MN: \( \frac{4}{3} \) 4. Length of MN: \( 5 \) 5. Area of Triangle ABC: \( 34 \) Feel free to ask if you have any other questions or need further assistance!