The problem of parallel lines and inflection points is one of the common questions in high school mathematics, and its solution is both regular and enlightening. Starting with ten typical examples, this paper analyzes and explains parallel lines and inflection points.

Model 1: Determine whether the line segments are parallel.

This question gives two line segments, which are required to judge whether they are parallel. Generally speaking, it needs to be calculated according to the slope of the straight line where the line segment is located. Two lines are parallel if their slopes are equal, otherwise they are not parallel. This question is relatively simple, so I won't go into details.

Model 2: Line Projection Comparison

This question requires giving the projection and orientation of line segments in the coordinate system and comparing the projection sizes of different line segments. It is necessary to compare the projection sizes of line segments in the horizontal or vertical direction according to their orientations. It should be noted that if two line segments are in opposite directions, it is necessary to compare their absolute projection values.

Mode 3: Translation constitutes parallel lines.

According to the existing line segments and vectors, this question is translated into a parallel line segment. It should be noted that the length of the translation line segment is equal to the length of the original line segment, but the direction is also parallel to the direction of the original line segment. Therefore, you only need to apply the vector to one point to get another point, and these two points can be connected to form the required parallel line segment.

Mode 4: inflection point judgment

This question gives two line segments, and it is required to judge whether there is an intersection. At this time, it needs to be discussed in two situations: first, the straight lines where the two line segments are located are parallel, and then if the two line segments have something in common, there will be intersections; Second, the straight lines of the two line segments are not parallel, and it is only necessary to judge whether the straight lines of the two line segments intersect.

Model 5: Judging the inflection point position from the angle.

This question type requires that two line segments be constructed with a certain point as the inflection point, so that the angle formed by the two line segments is fixed. At this time, it should be noted that only when the included angle of two line segments is acute angle/right angle will the two line segments have common points; When the included angle between two line segments is obtuse, there will be no common point between the two line segments.

Mode 6: Whether the angles are parallel.

This question gives the included angle between two line segments, which requires judging whether they are parallel. At this time, it is necessary to use sine theorem to substitute the length and included angle of the line segment into the calculation and get the slope of the two line segments. If the slopes are the same, they are parallel.

Model 7: Find the coordinates of the intersection of two line segments.

This question gives two line segments, and it is required to find the intersection coordinates of the two line segments. It is necessary to solve the intersection coordinates by determinant method according to the analytical formula of the straight line where the two line segments are located. It should be noted that when the straight lines of two line segments are parallel, the two line segments have no intersection.

Model 8: Find the intersection coordinates of two line segments after translation.

This question type is similar to model 7, but it needs to translate a line segment first, and then find the intersection. It should be noted that the length and included angle between the translated line segment and the original line segment remain unchanged, which increases the difficulty of solving.

Model 9: Solution of Angle Boundary Line of Pliers

This problem requires that a triangle be constructed between two parallel lines, and that the inner angle of the triangle be a pincer angle or an outer angle. At this time, it is necessary to use sine/cosine theorem to solve the unknown angle or line segment length according to the relationship between line segment length and included angle.

Mode 10: Translation constructs right angles.

This question requires that a line segment and a vector form a right angle. At this time, you can take the vector as the normal vector of the line segment, adjust the length of the line segment according to the module length of the vector, and then translate it.

abstract

In the problem of parallel lines and inflection points, different problems have different solutions, but they can be summarized as follows: comparing by calculating slopes, constructing parallel lines by translation, and solving them by intersection points. Therefore, in solving problems, we need to have flexible thinking and strict logical reasoning ability in order to master this kind of questions more skillfully.