Junior high School Math--(not in class, but very simple knowledge)

Memories of Junior High school mathematics, only to find themselves and learn something very limited, a lot of things in junior high School is simply involved. Recently returned to junior high school knowledge, summed up some of the mathematical self-discovery and simple reasoning things.

First, the Pythagorean theorem

Many people know that the Pythagorean theorem, also can be successfully proved out, after all, the textbook provides a lot of the method of quadrilateral proof, but if give a circle and a right triangle, can prove it? The answer is yes, this is believed that many people in junior high school time has already discovered (I was studying the circle time accidentally discovers).

In this triangle, we set the radius of the circle to R, which is eo=do=fo=r,ac=a,bc=b,ab=c.

We all know that R can be expressed by a, B, C, using the area method can be easily obtained r=ab/(a+b+c),using the nature of the circle tangent can be easily obtained r= (A+B-C)/2.

The two equations are equal and can be rolled out a^2+b^2=c^2. And in the process of proving that there is no inference associated with the Pythagorean theorem or itself, so we can think of this as a correct way to prove it.

Second, PARABOLIC

(1), do not know how many people and I, in the middle school study has not been to understand the definition of parabola, then began to use. In high school, or the Xinhua dictionary tells me what is a parabola: the set of points that are equal to the point and the fixed line.

For a parabolic line perpendicular to the x axis, we can make some simple inferences.

Let's start with the reverse:

Then put the situation special point, set the parabolic analytic type Y=AX^2+C (a<>0), then we can take the fixed line as the x-axis, fixed-point coordinates M (0,2C). The point coordinates on the parabolic line are P (t,at^2+c), the S1 is the distance from the point p to the X axis, and the S2 is the length of the segment PM. S1=at^2+c,

s2= (t^2+ (2c-at^2-c) ^2) ^ (1/2), the process of simplification in the middle is not written. S1^2=a^2t^4+c^2+2act^2,s2^2= (1-2AC) t^2+a^2t^4+c^2, compare S1 and S2 we can see (1-2AC) t^2=2act^2∴c=1/(4a)

so y=ax^2+ (1/(4a)), so the coordinates of the fixed point is (0,1/(2a)), the fixed line is the X axis.

By Translating this parabola, you can get the fixed point (-b/(2a), (4ac-b^2+1)/(4a)) of any parabolic y=ax^2+bx+c (A<>0), and the analytic formula for the definite line is y= (4ac-b^2-1 ) /(4a)).

We go back to the original definition, for a fixed point (p,q) and a fixed line y=t (here, in order to simplify, we only see the line parallel to the x-axis case), we can go to the parabola of the analytic formula is y= (1/(2Q-2T)) x^2+ (p/(t-q)) x+ ((p^2+q^2-t^ 2)/(2Q-2T)).

Thus, for the axis of symmetry perpendicular to the y-axis can also be the X, Y interchange, of course, the axis of the y-axis and the parabolic symmetry axis parallel, and then do some rotation can be, although Y is no longer the function of X, but the relationship between X and Y can also be expressed.

For a parabola, the light (a straight line in mathematics) that passes through the focal point (that is, the previously stated point) must be parallel to the symmetric axis after the parabolic reflection.

(2), application: Finally, we look at a magical physics of the application of the parabola (middle school when the weak and weak do not know).

As we all know, the bicycle lamp with each other perpendicular to the two flat mirror, it will be able to shoot the incident light parallel. For some headlights, their lampshade is obviously not two perpendicular to each other flat mirror so simple, there is a parabola around the axis of rotation of the spatial figure, we use the above properties can be found that this light scattering less, also can make light more bright concentration.

The proof is as follows: in parabolic y=ax^2+ (1/(4a)) (a<>0), take the point P (t,at^2+ (1/(4a)), we from this point to the X axis of vertical, perpendicular C, parabolic focus m (0,1/(2a)), p points at the intersection of the parabola tangent and the y axis is D , then the analytic formula of this tangent line y=2atx+1/(4a)-at^2 (the linear analytic type K value is the slope of the parabola at p Point), then the d point coordinate is (0,1/(4a)-at^2), md=mp=pc, and md∥pc, so the quadrilateral MDCP is a diamond, Thus, the parallel of the normal (i.e. the perpendicular of the tangent of the M-point) over the M-point is divided by the angular dmp so that the angle between the reflected light and the incident light is equal to the angular DMP, thus proving that the reflected light is parallel to the y-axis.

Hope to discover more beautiful mathematics in the future, to enrich the small blog.

Junior high School Math--(not in class, but very simple knowledge)