Let's break down the problem into two main parts as requested:
### Part (i): Show that [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] bisect each other at right angles.
1. Midpoints of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex]:
- Midpoint of [tex]\(AC\)[/tex]:
To find the midpoint of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
For [tex]\(A(4, 0)\)[/tex] and [tex]\(C(-10, 6)\)[/tex]:
[tex]\[
\text{Midpoint of } AC = \left( \frac{4 + (-10)}{2}, \frac{0 + 6}{2} \right) = \left( \frac{-6}{2}, \frac{6}{2} \right) = (-3.0, 3.0)
\][/tex]
- Midpoint of [tex]\(BD\)[/tex]:
For [tex]\(B(14, 11)\)[/tex] and [tex]\(D(-10, -5)\)[/tex]:
[tex]\[
\text{Midpoint of } BD = \left( \frac{14 + (-10)}{2}, \frac{11 + (-5)}{2} \right) = \left( \frac{4}{2}, \frac{6}{2} \right) = (2.0, 3.0)
\][/tex]
The midpoints are [tex]\((-3.0, 3.0)\)[/tex] and [tex]\((2.0, 3.0)\)[/tex], indicating they do not coincide.
Since the midpoints [tex]\(({-3.0, 3.0})\)[/tex] and [tex]\((2.0, 3.0)\)[/tex] are not the same, [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] do not bisect each other.
2. Slopes of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex]:
- Slope of [tex]\(AC\)[/tex]:
The slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For [tex]\(A(4, 0)\)[/tex] and [tex]\(C(-10, 6)\)[/tex]:
[tex]\[
\text{Slope of } AC = \frac{6 - 0}{-10 - 4} = \frac{6}{-14} = -\frac{3}{7} = -0.42857142857142855
\][/tex]
- Slope of [tex]\(BD\)[/tex]:
For [tex]\(B(14, 11)\)[/tex] and [tex]\(D(-10, -5)\)[/tex]:
[tex]\[
\text{Slope of } BD = \frac{-5 - 11}{-10 - 14} = \frac{-16}{-24} = \frac{2}{3} = 0.6666666666666666
\][/tex]
3. Perpendicularity:
The product of the slopes of two perpendicular lines is [tex]\(-1\)[/tex]. Calculate the product of the slopes:
[tex]\[
\text{Product of slopes} = (-0.42857142857142855) \times 0.6666666666666666 \approx -0.2857142857142857
\][/tex]
Since the product is not [tex]\(-1\)[/tex], [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are not perpendicular.
### Part (ii): Calculate the ratio [tex]\(|BD| : |AC|\)[/tex].
1. Length of [tex]\(BD\)[/tex]:
The length of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
For [tex]\(B(14, 11)\)[/tex] and [tex]\(D(-10, -5)\)[/tex]:
[tex]\[
\text{Length of } BD = \sqrt{(14 - (-10))^2 + (11 - (-5))^2} = \sqrt{(24)^2 + (16)^2} = \sqrt{576 + 256} = \sqrt{832} \approx 28.844410203711913
\][/tex]
2. Length of [tex]\(AC\)[/tex]:
For [tex]\(A(4, 0)\)[/tex] and [tex]\(C(-10, 6)\)[/tex]:
[tex]\[
\text{Length of } AC = \sqrt{(4 - (-10))^2 + (0 - 6)^2} = \sqrt{(14)^2 + (-6)^2} = \sqrt{196 + 36} = \sqrt{232} \approx 15.231546211727817
\][/tex]
3. Ratio [tex]\(|BD| : |AC|\)[/tex]:
[tex]\[
\text{Ratio} = \frac{|BD|}{|AC|} = \frac{28.844410203711913}{15.231546211727817} \approx 1.8937283058959973
\][/tex]
### Summary:
- [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] do not bisect each other.
- [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are not perpendicular.
- The ratio [tex]\(|BD| : |AC|\)[/tex] is approximately [tex]\(1.8937\)[/tex].
