Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Sure! Let's tackle each part of the question step by step.
### Given Points
We have two points:
- Point A: (2, 5)
- Point B: (-1, 7)
### Part (a) - Calculate the gradient of the interval which joins them.
The gradient (or slope) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the following formula:
[tex]\[ \text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For our given points:
- [tex]\(x_1 = 2\)[/tex], [tex]\(y_1 = 5\)[/tex]
- [tex]\(x_2 = -1\)[/tex], [tex]\(y_2 = 7\)[/tex]
Plugging in these values, we get:
[tex]\[ \text{Gradient} = \frac{7 - 5}{-1 - 2} = \frac{2}{-3} = -\frac{2}{3} \][/tex]
So, the gradient of the interval which joins the points (2, 5) and (-1, 7) is [tex]\(-0.6666666666666666\)[/tex] (or [tex]\(-\frac{2}{3}\)[/tex]).
### Part (b) - Calculate the distance between them.
The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the values, we obtain:
[tex]\[ \text{Distance} = \sqrt{(-1 - 2)^2 + (7 - 5)^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \][/tex]
Thus, the distance between the points (2, 5) and (-1, 7) is approximately [tex]\(3.605551275463989\)[/tex], when evaluated.
### Part (c) - Calculate the midpoint of the interval which joins them.
The midpoint [tex]\((x_m, y_m)\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the midpoint formula:
[tex]\[ (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \][/tex]
Plugging in the values, we get:
[tex]\[ (x_m, y_m) = \left(\frac{2 + (-1)}{2}, \frac{5 + 7}{2}\right) = \left(\frac{1}{2}, \frac{12}{2}\right) = (0.5, 6) \][/tex]
So, the midpoint of the interval which joins the points (2, 5) and (-1, 7) is [tex]\((0.5, 6)\)[/tex].
### Summary
1. The gradient of the interval joining the points is [tex]\(-0.6666666666666666\)[/tex].
2. The distance between the points is approximately [tex]\(3.605551275463989\)[/tex].
3. The midpoint of the interval joining the points is [tex]\((0.5, 6)\)[/tex].
### Given Points
We have two points:
- Point A: (2, 5)
- Point B: (-1, 7)
### Part (a) - Calculate the gradient of the interval which joins them.
The gradient (or slope) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the following formula:
[tex]\[ \text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For our given points:
- [tex]\(x_1 = 2\)[/tex], [tex]\(y_1 = 5\)[/tex]
- [tex]\(x_2 = -1\)[/tex], [tex]\(y_2 = 7\)[/tex]
Plugging in these values, we get:
[tex]\[ \text{Gradient} = \frac{7 - 5}{-1 - 2} = \frac{2}{-3} = -\frac{2}{3} \][/tex]
So, the gradient of the interval which joins the points (2, 5) and (-1, 7) is [tex]\(-0.6666666666666666\)[/tex] (or [tex]\(-\frac{2}{3}\)[/tex]).
### Part (b) - Calculate the distance between them.
The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the values, we obtain:
[tex]\[ \text{Distance} = \sqrt{(-1 - 2)^2 + (7 - 5)^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \][/tex]
Thus, the distance between the points (2, 5) and (-1, 7) is approximately [tex]\(3.605551275463989\)[/tex], when evaluated.
### Part (c) - Calculate the midpoint of the interval which joins them.
The midpoint [tex]\((x_m, y_m)\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the midpoint formula:
[tex]\[ (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \][/tex]
Plugging in the values, we get:
[tex]\[ (x_m, y_m) = \left(\frac{2 + (-1)}{2}, \frac{5 + 7}{2}\right) = \left(\frac{1}{2}, \frac{12}{2}\right) = (0.5, 6) \][/tex]
So, the midpoint of the interval which joins the points (2, 5) and (-1, 7) is [tex]\((0.5, 6)\)[/tex].
### Summary
1. The gradient of the interval joining the points is [tex]\(-0.6666666666666666\)[/tex].
2. The distance between the points is approximately [tex]\(3.605551275463989\)[/tex].
3. The midpoint of the interval joining the points is [tex]\((0.5, 6)\)[/tex].
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
