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…
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
- Length of AB: \( 2\sqrt{13} \)
- Midpoint of AB: \( (-4, 4) \)
- Slope of MN: \( \frac{4}{3} \)
- Length of MN: \( 5 \)
- Area of Triangle ABC: \( 34 \)
Feel free to ask if you have any other questions or need further assistance!
