Let's calculate the distance between the given pair of points using the distance formula, which is derived from the Pythagorean theorem. The distance formula to find the distance [tex]\( D \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a 2-dimensional space is:
[tex]\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Given the pair of points in question (i):
[tex]\((b+c, c+a)\)[/tex] and [tex]\((c+a, a+b)\)[/tex]
Let's denote these points with coordinates:
- [tex]\((x_1, y_1) = (b+c, c+a)\)[/tex]
- [tex]\((x_2, y_2) = (c+a, a+b)\)[/tex]
1. First, we need to find the differences in the x-coordinates and y-coordinates:
[tex]\[
x_2 - x_1 = (c + a) - (b + c)
\][/tex]
[tex]\[
y_2 - y_1 = (a + b) - (c + a)
\][/tex]
Simplifying these differences:
[tex]\[
x_2 - x_1 = c + a - b - c = a - b
\][/tex]
[tex]\[
y_2 - y_1 = a + b - c - a = b - c
\][/tex]
2. Next, we substitute these values into the distance formula:
[tex]\[
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting [tex]\( x_2 - x_1 \)[/tex] and [tex]\( y_2 - y_1 \)[/tex] we found:
[tex]\[
D = \sqrt{(a - b)^2 + (b - c)^2}
\][/tex]
3. Evaluating the above formula:
[tex]\[
D = \sqrt{(a - b)^2 + (b - c)^2}
\][/tex]
For the sake of simplicity, let’s assume [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 3 \)[/tex] as per the example values:
- [tex]\((x_1, y_1) = (2 + 3, 3 + 1) = (5, 4)\)[/tex]
- [tex]\((x_2, y_2) = (3 + 1, 1 + 2) = (4, 3)\)[/tex]
Calculating the differences:
[tex]\[
x_2 - x_1 = 4 - 5 = -1
\][/tex]
[tex]\[
y_2 - y_1 = 3 - 4 = -1
\][/tex]
Applying these differences to the distance formula:
[tex]\[
D = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4142
\][/tex]
Therefore, the distance between the points [tex]\((b+c, c+a)\)[/tex] and [tex]\((c+a, a+b)\)[/tex] for the chosen values is approximately [tex]\( \sqrt{2} \)[/tex] or 1.4142.
For part (ii), it seems incomplete. Please provide the full expression or question to proceed with the solution.
