Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Let's analyze the given quadrilateral [tex]\(ABCD\)[/tex] with vertices [tex]\(A(4,0)\)[/tex], [tex]\(B(14,11)\)[/tex], [tex]\(C(-10,6)\)[/tex], and [tex]\(D(-10,-5)\)[/tex].
### Part (i): Show that [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] bisect each other at right angles.
To show that [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] bisect each other, we need to find the midpoints of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] and verify that they coincide. Then, we need to verify that the slopes of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are negative reciprocals, indicating they are perpendicular.
#### Midpoints Calculation
The midpoint of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
For segment [tex]\(AC\)[/tex]:
- Coordinates of [tex]\(A\)[/tex]: [tex]\( (4, 0) \)[/tex]
- Coordinates of [tex]\(C\)[/tex]: [tex]\((-10, 6)\)[/tex]
[tex]\[ \text{Midpoint of } AC = \left( \frac{4 + (-10)}{2}, \frac{0 + 6}{2} \right) = \left( \frac{-6}{2}, \frac{6}{2} \right) = (-3, 3) \][/tex]
For segment [tex]\(BD\)[/tex]:
- Coordinates of [tex]\(B\)[/tex]: [tex]\( (14, 11) \)[/tex]
- Coordinates of [tex]\(D\)[/tex]: [tex]\((-10, -5)\)[/tex]
[tex]\[ \text{Midpoint of } BD = \left( \frac{14 + (-10)}{2}, \frac{11 + (-5)}{2} \right) = \left( \frac{4}{2}, \frac{6}{2} \right) = (2, 3) \][/tex]
Since the midpoints of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are [tex]\((-3, 3)\)[/tex] and [tex]\( (2, 3)\)[/tex], respectively, they do not coincide, so [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] do not bisect each other at the same midpoint. Hence, [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] do not bisect each other.
#### Slopes Calculation
The slope [tex]\(m\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For segment [tex]\(AC\)[/tex]:
[tex]\[ \text{Slope of } AC = \frac{6 - 0}{-10 - 4} = \frac{6}{-14} = -\frac{3}{7} = -0.42857142857142855 \][/tex]
For segment [tex]\(BD\)[/tex]:
[tex]\[ \text{Slope of } BD = \frac{-5 - 11}{-10 - 14} = \frac{-16}{-24} = \frac{2}{3} = 0.6666666666666666 \][/tex]
The product of the slopes:
[tex]\[ (-0.42857142857142855) \times (0.6666666666666666) = -0.2857142857142857 \][/tex]
Since the product of the slopes is not -1, [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are not perpendicular. Thus, they do not bisect each other at right angles.
### Part (ii): Calculate the ratio [tex]\(\lvert BD \rvert : \lvert AC \rvert \)[/tex].
#### Lengths Calculation
The length of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For segment [tex]\(AC\)[/tex]:
[tex]\[ \lvert AC \rvert = \sqrt{(-10 - 4)^2 + (6 - 0)^2} = \sqrt{(-14)^2 + (6)^2} = \sqrt{196 + 36} = \sqrt{232} = 15.231546211727817 \][/tex]
For segment [tex]\(BD\)[/tex]:
[tex]\[ \lvert BD \rvert = \sqrt{(-10 - 14)^2 + (-5 - 11)^2} = \sqrt{(-24)^2 + (-16)^2} = \sqrt{576 + 256} = \sqrt{832} = 28.844410203711913 \][/tex]
#### Ratio Calculation
The ratio [tex]\(\frac{\lvert BD \rvert}{\lvert AC \rvert}\)[/tex] is therefore:
[tex]\[ \text{Ratio} = \frac{28.844410203711913}{15.231546211727817} = 1.8937283058959973 \][/tex]
So the ratio of [tex]\(\lvert BD \rvert\)[/tex] to [tex]\(\lvert AC \rvert\)[/tex] is approximately [tex]\(1.8937\)[/tex].
### Part (i): Show that [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] bisect each other at right angles.
To show that [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] bisect each other, we need to find the midpoints of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] and verify that they coincide. Then, we need to verify that the slopes of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are negative reciprocals, indicating they are perpendicular.
#### Midpoints Calculation
The midpoint of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
For segment [tex]\(AC\)[/tex]:
- Coordinates of [tex]\(A\)[/tex]: [tex]\( (4, 0) \)[/tex]
- Coordinates of [tex]\(C\)[/tex]: [tex]\((-10, 6)\)[/tex]
[tex]\[ \text{Midpoint of } AC = \left( \frac{4 + (-10)}{2}, \frac{0 + 6}{2} \right) = \left( \frac{-6}{2}, \frac{6}{2} \right) = (-3, 3) \][/tex]
For segment [tex]\(BD\)[/tex]:
- Coordinates of [tex]\(B\)[/tex]: [tex]\( (14, 11) \)[/tex]
- Coordinates of [tex]\(D\)[/tex]: [tex]\((-10, -5)\)[/tex]
[tex]\[ \text{Midpoint of } BD = \left( \frac{14 + (-10)}{2}, \frac{11 + (-5)}{2} \right) = \left( \frac{4}{2}, \frac{6}{2} \right) = (2, 3) \][/tex]
Since the midpoints of [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are [tex]\((-3, 3)\)[/tex] and [tex]\( (2, 3)\)[/tex], respectively, they do not coincide, so [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] do not bisect each other at the same midpoint. Hence, [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] do not bisect each other.
#### Slopes Calculation
The slope [tex]\(m\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For segment [tex]\(AC\)[/tex]:
[tex]\[ \text{Slope of } AC = \frac{6 - 0}{-10 - 4} = \frac{6}{-14} = -\frac{3}{7} = -0.42857142857142855 \][/tex]
For segment [tex]\(BD\)[/tex]:
[tex]\[ \text{Slope of } BD = \frac{-5 - 11}{-10 - 14} = \frac{-16}{-24} = \frac{2}{3} = 0.6666666666666666 \][/tex]
The product of the slopes:
[tex]\[ (-0.42857142857142855) \times (0.6666666666666666) = -0.2857142857142857 \][/tex]
Since the product of the slopes is not -1, [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are not perpendicular. Thus, they do not bisect each other at right angles.
### Part (ii): Calculate the ratio [tex]\(\lvert BD \rvert : \lvert AC \rvert \)[/tex].
#### Lengths Calculation
The length of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For segment [tex]\(AC\)[/tex]:
[tex]\[ \lvert AC \rvert = \sqrt{(-10 - 4)^2 + (6 - 0)^2} = \sqrt{(-14)^2 + (6)^2} = \sqrt{196 + 36} = \sqrt{232} = 15.231546211727817 \][/tex]
For segment [tex]\(BD\)[/tex]:
[tex]\[ \lvert BD \rvert = \sqrt{(-10 - 14)^2 + (-5 - 11)^2} = \sqrt{(-24)^2 + (-16)^2} = \sqrt{576 + 256} = \sqrt{832} = 28.844410203711913 \][/tex]
#### Ratio Calculation
The ratio [tex]\(\frac{\lvert BD \rvert}{\lvert AC \rvert}\)[/tex] is therefore:
[tex]\[ \text{Ratio} = \frac{28.844410203711913}{15.231546211727817} = 1.8937283058959973 \][/tex]
So the ratio of [tex]\(\lvert BD \rvert\)[/tex] to [tex]\(\lvert AC \rvert\)[/tex] is approximately [tex]\(1.8937\)[/tex].
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.