Example 1. (Qianjiang, Hubei, 2007) As shown in figure 1, there are two villages A and B along the river. A waterworks will be built along the river to supply water to villages A and B. 。
(1) If the distance between the factory and village A and village B is equal, where should we choose to build the factory?
(2) Where should the water pipes from the factory to villages A and B be built to save the most materials?
The analysis shows that the distance from (1) to point A and point B is equal, which can be associated with "the distance from the point on the vertical line in the line segment is equal to both ends of the line segment".
(2) To make the sum of the distances from the factory to villages A and B shortest, we can think of "the shortest line segment between two points".
Solution: (1) As shown in Figure 2, take the midpoint G of the line segment AB, draw the vertical line of AB through the midpoint G, and pass through EF and P, then the distance from P to A and B is equal.
(2) As shown in Figure 3, draw the symmetrical point A' of point A relative to the bank EF, and connect point A'b with EF at point P, then the sum of the distances from p to AB is the shortest.
Comments: If we pay attention, many people make use of axial symmetry in our lives. If we observe and think more at ordinary times, we will find that axial symmetry can also help us solve problems.
Example 2. As shown in Figure 3, two roads OA and OB intersect, and there is an oil depot in the middle of the two roads, which is set as point P. If there are gas stations on each road, please design a plan where to set up two gas stations, so that the tanker can take the shortest route from the oil depot, pass through one gas station, then go to another gas station and finally return to the oil depot.
The analysis shows that this is a practical problem, and we need to turn it into a mathematical problem. After analysis, we know that this problem is to find the shortest distance for an oil tanker, OA and OB intersect, and point P is within ∠AOB. Usually, we think of symmetry, and make the symmetrical points P 1 and P2 of point P about straight lines OA and OB respectively, and connect P 1P2 to OA and OB of point C respectively. We can use the trilateral relationship of a triangle to illustrate this.
Solution: Make the symmetrical points of point P about straight lines OA and OB, P 1, P2,
Connect P 1P2 to c, d,
Then C and D are the locations where gas stations are built.
If we take a point different from c and d,
According to the triangle relationship, the shortest distance is to build a gas station tanker at point C and D respectively.
Comments: There is no detailed explanation here why it takes the shortest distance to build gas station trucks at C and D. Please think about it and find out.
Example 3. (Jingmen, Hubei, 2007) A water pump station will be built along the river to supply water to Village A and Village B respectively. The pump station should be built by the river. Where can the water pipes be used the shortest?
Analyze and solve this problem, find out the symmetrical point of point A about the straight line, and connect the intersecting straight line with point P, then point P is the position of the point with the shortest sum of distances to village A and village B.
The reason can be known from the nature of axial symmetry.
If you choose another point (different from P), add
Yes,
that is
Therefore, it is the shortest.
It can be seen that axial symmetry helps us find the position of the point that meets the requirements.
Comments: The solution of this problem provides us with a clue to solve the problem, which leads to a series of problem-solving ideas. Let students feel the natural presentation of knowledge and experience the mystery and fun of mathematics in the process of operating activities.
The combination of numbers and shapes in the shortest distance
-Talking about Mathematics Question 20 of Enshi Senior High School Entrance Examination in 2008
This topic uses the idea of combining numbers and shapes to comprehensively examine the application ability of students' knowledge of geometry and algebra. From the way of communication, the first question requires students to combine special algebra with the evaluation of shape by using the characteristics of shape, first get the law by exploring the guiding form, and then use the geometric knowledge "the shortest line segment between two points" to find the minimum value of algebra.
The whole process fully shows the general process of students learning new mathematics knowledge: cognition-demonstration-application. It is a successful example of mathematical communication.
The first quiz is designed to familiarize students with the relationship between this special algebraic expression and graphics, find out the "expression" contained in "form", and have certain observation and association ability;
The second test was designed as an investigation process. In the case that "form and form" are already available, it is a test of students' study habits and requires students to have the ability of autonomous learning.
The design of the third question is mainly to apply and expand the conclusions explored.
The whole process embodies the general law in special problems and is a natural regression of mathematical knowledge and problem-solving methods.
Examples are as follows:
As shown in the figure, C is a moving point on the line segment BD, passing through points B and D of AB ⊥ BD and ED ⊥ BD respectively, connecting AC and EC. It is known that AB=5, DE= 1, BD=8 and CD = X.
The length of (1)AC+Ce is expressed by an algebraic expression containing x;
(2) What conditions does point C meet, and the value of AC+CE is the smallest?