In a Cartesian coordinate system, two points a(2,3) and b(-2,-1) are given. Find the equation for the perpendicular line of the straight line ab.
Ideas for solving the problem. To solve this problem, we need to apply knowledge of the properties of the perpendicular line, the slope of the straight line, and the equation of the point-oblique line. We divide the problem-solving process into the following steps:
Step 1: Understand the definition of the perpendicular line.
A perpendicular line is a midpoint that connects two point segments and is perpendicular to the line segment. Our first step is to find the midpoint of the line segment AB and make sure that it is perpendicular to the AB segment.
Step 2: Calculate the midpoint of line segment AB.
The midpoint M of the line segment AB can be obtained by calculating the average of the coordinates of points A and B. The coordinates of the midpoint m are:
mx=frac
my=frac
Substituting the coordinates of a(2,3) and b(-2,-1), we get:
mx=frac=0
my=frac=1
The coordinates of the midpoint m are (0,1).
Step 3: Calculate the slope of the line ab.
The slope k of the straight line can be calculated by the following formula: k=fracFor the straight line ab we use the coordinates of the points a and b to calculate the slope:
k=frac=frac=1。
The slope of the straight line ab is 1.
Step 4: Determine the slope of the perpendicular.
Since the perpendicular line is perpendicular to the ab line, the product of the slope of the perpendicular line and the slope of the ab line should be -1. If the slope of the straight line ab is k, the slope of the perpendicular line is k'It will be:.
k'=-frac}
Substituting k = 1, we get k'=-1。
Step 5: Use the point-oblique equation to determine the perpendicular equation.
Now that we have a point of the perpendicular line (the midpoint m(0,1)) and a slope (-1), we can use the point oblique equation to write the equation for the perpendicular. The form of the point-oblique equation is:
yy1=m(xx1) is substituted for m(0,1) and the slope -1, and we get the equation for the perpendicular:
y1=-1(x0)
Simplifying this equation we get:
y=-x1: From the above steps, we get the equation of the vertical line of the straight line ab, which is y=-x1. This process not only shows how to find the middle perpendicular line of a line segment, but also reviews the relevant knowledge of slope and point-oblique equations, which is a good comprehensive application of geometric and algebraic knowledge.