2.1 Mid-point and Length of a Line Segment
1. What are coordinates of a point in the Cartesian plane?
The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These lines are perpendicular and intersect at the origin, forming four quadrants.
A point in the Cartesian plane is represented by an ordered pair of numbers \((x,y)\), where ‘x’ is the distance from the point to the y-axis (x-coordinate), and ‘y’ is the distance from the point to the x-axis (y-coordinate).
2. What is the midpoint of a line segment?
The midpoint of a line segment is the point that divides the line segment into two equal segments. It is exactly halfway between the endpoints of the line segment.
3. How to find the midpoint?
The midpoint \(M(x,y)\) of a line segment with endpoints \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the Cartesian plane can be found using the following formulas:
\[x = \frac{{x_1+x_2}}{2}\] \[y = \frac{{y_1+y_2}}{2}\]4. How to find coordinates of one of the endpoints given the midpoint?
If you know the coordinates of the midpoint \(M(x,y)\) and one endpoint \(A(x_1,y_1)\), you can find the coordinates of the other endpoint \(B(x_2,y_2)\) using the following formulas:
\[x_2= 2x - x_1\] \[y_2= 2y - y_1\]5. What is the distance formula?
The distance formula is used to determine the distance, \(d\), between two points in the Cartesian plane. If the points are \(A(x_1,y_1)\) and \(B(x_2,y_2)\), the distance between them is given by:
\[d = \sqrt{{(x_2- x_1 )^2+ (y_2- y_1 )^2}}\]This formula is derived from the Pythagorean theorem and represents the length of the hypotenuse of a right triangle (the line segment between the two points).
6. How to find the length of a line segment?
The length of a line segment is the distance between its endpoints. So, you can use the distance formula to find the length of a line segment. If the endpoints of the line segment are \(A(x_1,y_1)\) and \(B(x_2,y_2)\), the length of the line segment is:
\[length = \sqrt{{(x_2- x_1 )^2+ (y_2- y_1 )^2}}\]