(1) The product of the slope of the straight line AB and the slope of the straight line L is-1, so as to establish the analytical formula of the straight line AB, and thus express the coordinates of the point B with the resolution function;
② Because the distances between point A and point B and straight line L are equal, point B can be found by using the formula of distance from point to straight line.
Two points are symmetrical about one point-if point A and point B are symmetrical about point O, and the coordinates of point A and point O are known, find the coordinates of point B:
(1) If two points are symmetrical about one point, that is, the three-point * * * line, the slope of the straight line AO is equal to the slope of the straight line bo, so that the coordinates of the point b can be expressed;
② Because AO=BO, it can be solved by the formula of line segment length.
Three lines are symmetrical about a straight line-if the straight line L 1 and the straight line L2 are symmetrical about the straight line L, and the straight line L 1 and the straight line L are known, find the straight line L2:
1) If three straight lines are parallel and the slopes are equal, the analytical formula of L2 can be set, and then the straight line L2 can be obtained from the equidistant formula of two parallel straight lines;
2) If three straight lines are not parallel, that is, three straight lines intersect, let the analytical formula of straight line L2 be y=kx+b, then:
① Find the coordinates of the intersection o of straight lines;
② The included angle between line L 1 and line L = the included angle between line L and line L2, and the slope of line L2 can be obtained from the angle formula;
(3) Substituting the point O into the analytical expression of the straight line L2 can find out B, from which the analytical expression of the straight line L2 can be found out.
Four straight lines are symmetrical about the point-if the straight line L 1 and the straight line L2 are symmetrical about the point O, and the straight line L 1 and the point O are known, find the straight line L2:
① Let the analytical formula of line L2 be y=kx+b, and list an equation with equal distance from point O to line L 1 and L2;
② Take any point A on the straight line L 1, find out the symmetrical point A' of point A on L2 about point O, and list an equation of oil AO = A'O;
③ If the equations in ① and ② are solved simultaneously, a straight line L2 can be obtained.
Five circles are symmetrical about the straight line —— If the circle O 1 and the circle O2 are symmetrical about the straight line L, and the straight line O 1 and the straight line L are symmetrical, find the circle O2:
① Two circles are symmetrical about the straight line L, and the radii r of these two circles are equal;
② If two circles are symmetrical about the straight line L, and the distance between the circle O 1 and the circle O2 and the straight line L is equal, it is transformed into a point-to-point symmetry problem about the straight line, as shown in Figure 2, from which the center of the circle O2 can be found;
③ Knowing the radius and center of the circle, we can write the analytical formula of the circle O2.
Six straight lines are symmetrical about a circle-if the straight line L 1 and the straight line L2 are symmetrical about a circle, and the straight line L 1 and the circle O are known, find the straight line L2:
(1) Write the coordinates of the center o of the circle o;
(2) Thus, the problem is transformed into a problem that a straight line is symmetrical about a point, as shown in Figure 4.
Seven circles are symmetrical about the point-if the circle O 1 and the circle O2 are symmetrical about the point a, and the circle O 1 and the point a are known, find the circle O2:
① The radii r of two circles are equal;
② Thus, the problem is transformed into a point-to-point symmetric problem, and the solution method is shown in Figure 2.
Eight points are symmetrical about a circle-if point A and point B are symmetrical about circle O, and point A and circle O are known, find point B:
(1) Write the coordinates of the center o of the circle o;
② Thus, the problem is transformed into a point-to-point symmetric problem, and the solution method is shown in Figure 2.