Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Sure, let's solve the given problem step-by-step.
### Given Points:
[tex]\( A(-1, 5) \)[/tex] and [tex]\( B(4, -3) \)[/tex]
### 1. Distance Between Two Points
To find the distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ d = \sqrt{(4 - (-1))^2 + (-3 - 5)^2} \][/tex]
[tex]\[ d = \sqrt{(4 + 1)^2 + (-3 - 5)^2} \][/tex]
[tex]\[ d = \sqrt{5^2 + (-8)^2} \][/tex]
[tex]\[ d = \sqrt{25 + 64} \][/tex]
[tex]\[ d = \sqrt{89} \][/tex]
[tex]\[ d \approx 9.434 \][/tex]
So, the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is approximately [tex]\( 9.434 \)[/tex].
### 2. Midpoint of the Line Segment
The midpoint [tex]\( M \)[/tex] of a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ M = \left( \frac{-1 + 4}{2}, \frac{5 + (-3)}{2} \right) \][/tex]
[tex]\[ M = \left( \frac{3}{2}, \frac{2}{2} \right) \][/tex]
[tex]\[ M = \left( 1.5, 1.0 \right) \][/tex]
So, the midpoint of the line segment is [tex]\( (1.5, 1.0) \)[/tex].
### 3. Slope of the Line Segment
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ m = \frac{-3 - 5}{4 - (-1)} \][/tex]
[tex]\[ m = \frac{-3 - 5}{4 + 1} \][/tex]
[tex]\[ m = \frac{-8}{5} \][/tex]
[tex]\[ m = -1.6 \][/tex]
So, the slope of the line segment joining points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( -1.6 \)[/tex].
### Summary:
- The distance between points [tex]\( A(-1, 5) \)[/tex] and [tex]\( B(4, -3) \)[/tex] is approximately [tex]\( 9.434 \)[/tex].
- The midpoint of the line segment joining these points is [tex]\( (1.5, 1.0) \)[/tex].
- The slope of the line segment is [tex]\( -1.6 \)[/tex].
### Given Points:
[tex]\( A(-1, 5) \)[/tex] and [tex]\( B(4, -3) \)[/tex]
### 1. Distance Between Two Points
To find the distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ d = \sqrt{(4 - (-1))^2 + (-3 - 5)^2} \][/tex]
[tex]\[ d = \sqrt{(4 + 1)^2 + (-3 - 5)^2} \][/tex]
[tex]\[ d = \sqrt{5^2 + (-8)^2} \][/tex]
[tex]\[ d = \sqrt{25 + 64} \][/tex]
[tex]\[ d = \sqrt{89} \][/tex]
[tex]\[ d \approx 9.434 \][/tex]
So, the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is approximately [tex]\( 9.434 \)[/tex].
### 2. Midpoint of the Line Segment
The midpoint [tex]\( M \)[/tex] of a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ M = \left( \frac{-1 + 4}{2}, \frac{5 + (-3)}{2} \right) \][/tex]
[tex]\[ M = \left( \frac{3}{2}, \frac{2}{2} \right) \][/tex]
[tex]\[ M = \left( 1.5, 1.0 \right) \][/tex]
So, the midpoint of the line segment is [tex]\( (1.5, 1.0) \)[/tex].
### 3. Slope of the Line Segment
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ m = \frac{-3 - 5}{4 - (-1)} \][/tex]
[tex]\[ m = \frac{-3 - 5}{4 + 1} \][/tex]
[tex]\[ m = \frac{-8}{5} \][/tex]
[tex]\[ m = -1.6 \][/tex]
So, the slope of the line segment joining points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( -1.6 \)[/tex].
### Summary:
- The distance between points [tex]\( A(-1, 5) \)[/tex] and [tex]\( B(4, -3) \)[/tex] is approximately [tex]\( 9.434 \)[/tex].
- The midpoint of the line segment joining these points is [tex]\( (1.5, 1.0) \)[/tex].
- The slope of the line segment is [tex]\( -1.6 \)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.