Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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].
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.