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].
