A, B, C, And D Have The Coordinates \[$(-8,1), (-2,4), (-3,-1),\$\] And \[$(-6,5)\$\], Respectively. Which Sentence About The Points Is True?A. \[$A, B, C,\$\] And \[$D\$\] Lie On The Same Line.B.
Introduction
In mathematics, the concept of coordinates is a fundamental aspect of geometry and algebra. When dealing with points in a two-dimensional plane, their coordinates play a crucial role in determining their positions and relationships. In this article, we will explore the coordinates of points A, B, C, and D and determine which sentence about them is true.
The Coordinates
The coordinates of points A, B, C, and D are given as follows:
- A: (-8, 1)
- B: (-2, 4)
- C: (-3, -1)
- D: (-6, 5)
Understanding the Concept of Collinearity
Two or more points are said to be collinear if they lie on the same straight line. In other words, if we can draw a straight line that passes through all the points, then they are collinear. To determine whether points A, B, C, and D are collinear, we need to examine their coordinates and see if they satisfy the condition of collinearity.
Calculating the Slope
The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run). Mathematically, the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
We can use this formula to calculate the slope of the line passing through points A, B, C, and D.
Calculating the Slope for Points A and B
Let's calculate the slope of the line passing through points A and B.
m = (4 - 1) / (-2 - (-8)) = 3 / 6 = 1/2
Calculating the Slope for Points B and C
Now, let's calculate the slope of the line passing through points B and C.
m = (-1 - 4) / (-3 - (-2)) = -5 / -1 = 5
Calculating the Slope for Points C and D
Finally, let's calculate the slope of the line passing through points C and D.
m = (5 - (-1)) / (-6 - (-3)) = 6 / -3 = -2
Analyzing the Results
We have calculated the slopes of the lines passing through points A and B, B and C, and C and D. Now, let's analyze the results.
- The slope of the line passing through points A and B is 1/2.
- The slope of the line passing through points B and C is 5.
- The slope of the line passing through points C and D is -2.
As we can see, the slopes of the lines passing through points A and B, B and C, and C and D are not equal. This means that the lines passing through these points are not parallel, and therefore, points A, B, C, and D do not lie on the same line.
Conclusion
In conclusion, based on the coordinates of points A, B, C, and D, we can determine that they do not lie on the same line. The slopes of the lines passing through these points are not equal, which means that they are not parallel. Therefore, the correct sentence about the points is:
A, B, C, and D do not lie on the same line.
Final Answer
The final answer is:
Introduction
In our previous article, we explored the coordinates of points A, B, C, and D and determined that they do not lie on the same line. In this article, we will answer some frequently asked questions related to the coordinates of these points.
Q: What is the significance of the coordinates of points A, B, C, and D?
A: The coordinates of points A, B, C, and D are significant because they determine their positions in a two-dimensional plane. The coordinates (x, y) of a point represent its horizontal and vertical distances from a reference point, usually the origin (0, 0).
Q: How do you calculate the distance between two points?
A: To calculate the distance between two points (x1, y1) and (x2, y2), you can use the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
This formula calculates the distance between two points as the square root of the sum of the squares of the differences in their x-coordinates and y-coordinates.
Q: What is the concept of collinearity?
A: Collinearity is a concept in geometry where two or more points are said to be collinear if they lie on the same straight line. In other words, if we can draw a straight line that passes through all the points, then they are collinear.
Q: How do you determine if three points are collinear?
A: To determine if three points are collinear, you can calculate the slopes of the lines passing through each pair of points. If the slopes are equal, then the points are collinear.
Q: What is the significance of the slope of a line?
A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run). The slope of a line determines its direction and steepness.
Q: How do you calculate the slope of a line?
A: To calculate the slope of a line passing through two points (x1, y1) and (x2, y2), you can use the slope formula:
m = (y2 - y1) / (x2 - x1)
This formula calculates the slope of a line as the ratio of the vertical change to the horizontal change.
Q: What is the difference between parallel and perpendicular lines?
A: Parallel lines are lines that lie in the same plane and never intersect, while perpendicular lines are lines that intersect at a right angle (90 degrees). The slopes of parallel lines are equal, while the slopes of perpendicular lines are negative reciprocals of each other.
Q: How do you determine if two lines are parallel or perpendicular?
A: To determine if two lines are parallel or perpendicular, you can calculate their slopes. If the slopes are equal, then the lines are parallel. If the slopes are negative reciprocals of each other, then the lines are perpendicular.
Conclusion
In conclusion, the coordinates of points A, B, C, and D are significant because they determine their positions in a two-dimensional plane. The concept of collinearity is important in geometry, and the slope of a line determines its direction and steepness. We hope this Q&A article has provided you with a better understanding of the coordinates of points A, B, C, and D.
Final Answer
The final answer is:
- Q: What is the significance of the coordinates of points A, B, C, and D? A: The coordinates of points A, B, C, and D are significant because they determine their positions in a two-dimensional plane.
- Q: How do you calculate the distance between two points? A: To calculate the distance between two points (x1, y1) and (x2, y2), you can use the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
- Q: What is the concept of collinearity? A: Collinearity is a concept in geometry where two or more points are said to be collinear if they lie on the same straight line.
- Q: How do you determine if three points are collinear? A: To determine if three points are collinear, you can calculate the slopes of the lines passing through each pair of points. If the slopes are equal, then the points are collinear.
- Q: What is the significance of the slope of a line? A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run).
- Q: How do you calculate the slope of a line? A: To calculate the slope of a line passing through two points (x1, y1) and (x2, y2), you can use the slope formula: m = (y2 - y1) / (x2 - x1)
- Q: What is the difference between parallel and perpendicular lines? A: Parallel lines are lines that lie in the same plane and never intersect, while perpendicular lines are lines that intersect at a right angle (90 degrees).
- Q: How do you determine if two lines are parallel or perpendicular? A: To determine if two lines are parallel or perpendicular, you can calculate their slopes. If the slopes are equal, then the lines are parallel. If the slopes are negative reciprocals of each other, then the lines are perpendicular.