How exactly is the formula for the distance from a point to a straight line derived? Explain.

First of all, we must understand that the distance from the point to the straight line is the minimum distance between the point and the point on the straight line You can set a point as p(a, b), the linear line equation is y=kx ten d, you can take a point q(x, kx ten d) on the straight line, and the distance between pq is under the root number (a-x) 2 ten(b-kx-d) 2, and a quadratic function about x can be obtained by splitting it, and the formula can find its minimum value The formula process is too complicated, so the process is not given in the textbook

Let the coordinates of point a be (m,n), the equation of the line l is ax+by+c=0 on the line, when x=m, y=(-am-c) b, denoted as point b, denoted as point b, when y=n, x=(-bn-c) a, denoted as point c, obviously ab ac, through the point a as the ad straight line l at the point dad=ab*ac bc=|n+(am+c)/b|*|m+(bn+c)/a|Root number [(m+(bn+c) a) 2+(n+(am+c) b) 2].

After simplification, ad=|am+bn+c|Root number (A 2 + B 2).

First, find the slope of the straight line perpendicular to the known straight line, and then find the straight line equation perpendicular to the straight line through the coordinates of the known point, and find the intersection point by connecting the known straight line and the straight line equation, and the distance between the coordinates of the intersection point and the known coordinates can be found by using the Pythagorean theorem.

Point-to-straight distanceThe formula is ax+by+c=0. Straight line square Wu potato Hu Cheng.

is ax+by+c=0, and the coordinates of point p are (x0,y0). Of all the line segments connecting a point outside a line to a point on a line, the perpendicular segment is the shortest, and the length of this perpendicular segment is called the distance from the point to the line.

The point-to-straight line distance is the length of the perpendicular segment that connects a point outside the line with each point on the line.

Knowledge and skills of point-to-straight distances

Understand the derivation process of the point-to-straight line distance formula, and use the formula to find the distance from a fixed point to a fixed line, and understand the distance formula for two parallel straight lines.

Through the derivation of the distance formula from point to straight line, students can improve their understanding of the combination of numbers and shapes, deepen their awareness of using calculations to process figures, and transform the distance relationship between two parallel straight lines into a point-to-straight line distance.

The formula for the distance from a vector point to a line is:

Let the neutralization of the line l be ax+by+c=0, and the coordinates of the point p are (x0, y0), then the distance from the point p to the line l is:

In the same way, when p(x0,y0) and the analytic expression of the line l is y=kx+b, then the distance from the point p to the line l is:

Consider the point (x0, y0, z0) and the spatial line x-x1 l=y-y1 m=z-z1 n, with d=|(x1-x0，y1-y0，z1-z0)×(l，m，n)|/l²+m²+n²)。

Proof method. Think of the linear equation of the plane ax+by+c=0 as an equation of xyz space, which is a z-free equation, that is, an equation of a straight cylindrical surface (i.e., a plane).

Then find the distance from the point (x0,y0,0) to this plane (because it is the distance from (xy, y0) to ax+by+c=0, because this is equivalent to the distance from the point to the plane in the air at xy).

And according to the formula for the distance from the midpoint in space (x0, y0, z0) to the plane ax+by+cz+d=0:

d=|ax0+by0+cz0+d|/[a^2+b^2+c^2)]。

1. When the straight line is perpendicular to the x-axis.

From the properties of axisymmetric symmetry, the midpoint of y=b,aa' is on the line x=k, then,a+x) 2=k,x=2k-a

So it is easy to find the coordinates of a' (2k-a,b).

2. When the straight line is perpendicular to the y-axis.

From the properties of axisymmetry, it can be seen that the midpoint of x=a, bb' is on the line y=k, then, y+b) 2=k, y=2k-b

So it's easy to find the coordinates of b' (a,2k-b).

3. When the straight line is a general straight line, that is, its general form can be expressed as y=kx+b, which is transformed into the form of a straight line ax+by+c=0.

a, b) The coordinates of the symmetry point with respect to the line ax+by+c=0 are .

From the point of view of plane analytic geometry, a straight line on a plane is a graph represented by a binary linear equation in a plane Cartesian coordinate system.

The formula for the distance from a point to a straight line in high school math isd=│axo＋byo＋c│/√a²＋b²）。

Let the equation for the line l be ax+by+c=0 and the coordinates of the point p be (xo,yo), then the distance from the point p to the line l is:

d=│axo+byo+c│ /a²+b²）。

The point-to-straight distance is the length of the perpendicular segment that connects a point outside the line with each point on the line.

Formula Description:

The equation for the straight line in the formula is ax+by+c=0, and the coordinates of the point p are (x0, y0).

Among all the line segments connecting the outer point of the straight hock line and the points on the straight line, the perpendicular line segment is the shortest, and the length of this perpendicular line segment is called the distance from the point to the straight line.

Straight line ax+by+c=0 coordinates (xo,yo) then the distance from this point to this straight line is:

Formula description: The linear equation in the formula is ax+by+c=0, and the coordinates of the point p are (x0, y0).

Of all the line segments that connect a point outside a straight line with each point on a straight line, the perpendicular segment is the shortest, and the length of this perpendicular segment is called the distance from the point to the straight line.

If the line is ax by c 0 and the point coordinates are (xo, yo), then the distance from this point to the line is: axo byo c a b )

Process & Method Objectives:

1) Through the derivation of the distance formula from the point to the straight line, the students' understanding of the combination of numbers and shapes is improved, and the awareness of using "calculation" to deal with the lead simplification of "trouser shape in the figure" is deepened;

2) Convert the distance relationship between two parallel lines into a point-to-straight line distance.

According to the previous student, the formula for the distance from a point in space to a straight line is about the same as that of a plane, as follows:

Point (x0, y0, z0), linear equation ax+by+cz+d=0, distance formula: |a*x0+b*y0+c*z0+d|/sqrt(a^2+b^2+c^2)

Let the equation for the line with l be ax+by+c=0 and the coordinates of the point p are (xo,yo), then the distance from the point p to the line l is:

d=│axo+byo+c│ /a²+b²）。

The real trouble point above is p(x0,y0), the equation of the straight line ax by c=0 The formula for the distance from the point to the straight line d=|ax0 by0 c|[a 2 b 2)] a 2 b 2) indicates a squared plus b squared under the root number.

The distance from a point to a straight line can be obtained using the formula: |ax+by+c|The arithmetic square root of dividing by the square of a plus the square of b.

Take slippery selling as an example: find the distance from p(5,3) to the straight line 4x-3y+1=0.

Solution: l=|4*5-3*3+1|Divide by the square root of the arithmetic (16+9) =

Knowing that a point a(a,b) and a straight line l y=k1x+b1, and the straight line m y=k2x+b2 lets the straight line pass a and is perpendicular to the known straight line l, then k1*k2=-1, bring a into m, find m, and then connect l and m to find the intersection point b, and find the distance from a to l is the distance from point a to point b.

If you make a parallel line of a known straight line through the given point, then the distance from the point to the straight line is the distance between these two parallel lines.

Let the given point be (x0,y0), the straight line is ax+by+c=0, and the parallel line is (y-y0) (x-x0)=-a b

b(y-y0)+a(x-x0)=0

ax+by-(ax0+by0)=0

The distance between the two parallel lines is the difference of the constant term divided by the sum of the squares of the two coefficients of the root number, so it is ax0+by0+c root number (a 2 + b 2).

The concept of the straight-line distance from the point to the straight line is "a perpendicular line from a point outside the straight line, and the length of the line segment between the point before the letter and the perpendicular foot is called the distance from the point to the straight line of the straight line". So there is no situation that is not vertical.