Distance formula from point to line, what is the formula for calculating distance from point to line

Hello, I'm a freshman in high school, and the formula you asked us just taught this semester. The formula for the distance from the point po(xo,yo) to the straight line l:ax+by+c=0 is:

axo+byo+c=0|divide by the square of a + the sum of the squares of b and open the quadratic again; The point slope of a straight line with a point po(xo,yo) and a slope of k is: y-yo=k(x-xo) ; y=kx+b is oblique truncating, b is intercept; Slope k = tana (a is the degree of the angle of inclination).

If a(a,b).

b(c,d)

Then the distance from point A to point B is.

Turn left|Turn right.

Suppose we find c(a

b) the distance to the straight line l1: y=kx+h, because it is the distance from the point to the straight line, the point c should be used as the perpendicular line l2 of the straight line, then the obtained straight line l2 is perpendicular to y=kx+h, then the slope of the obtained straight line is -1 k (because the product of the slopes of two straight lines perpendicular to each other is -1, that is, k1*k2=-1), knowing the slope of this straight line and the point it wants to pass through, then the straight line l2 can be found, and then the intersection of l1 and l2 d can be found, Then it becomes to find the distance between point C and point D.

There's really no specific formula for this, but it's something to analyze.

There are formulas that can be applied when you analyze the situation and then when you need to get something.

If you know the coordinates of a point and the equation of a line.

Then you can do this point-to-line vertical line.

Then the vertical slope formula k1*k2=-1So the slope of the line equation (-1) = the slope of the perpendicular line

And then you know the equation for the two lines, and then you can figure out where they intersect.

Then the intersection point and the starting point are rooted (x1-x2) 2+(y1-y2) 2 to get the distance.

The formula is |aa+bb+c|Divide by a squared plus b squared under the root number.

The one below is the alpha of k=tan.

The formula for the distance from a point to a straight line is: d=(aa+bb+c) (a2+b2).

I really can't type it on the computer, see if you can understand it!

It's not very easy to play on the computer.

.Look at this is the slideshow, just click on it!

10, slide 10

axo+byo+c|Add square A under the root number divided by square B.

I'm a high 3, and I've always used this formula.

The distance from the point to the line is formulated as follows:

Let the equation for 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:

Definitional method proof:

By definition, the distance from the point p(x,y) to the line l:ax+by+c=0 is the perpendicular line from the point p to the line l.

The length of the segment. Let the perpendicular line from point p to the line be l', the vertical foot is q, then l'The slope of is b a then l'The analytic formula is y-y = (b a) (Hengbi elimination x-x).

Put l and l'Jointly instructed to know the l and l'The coordinates of the intersection point q are (Hui Nai (b 2x -aby -ac) (a 2+b 2), (a 2y -abx -bc) (a 2+b 2)) by the formula for the distance between the two points.

Pq 2=[(B 2X -Aby -AC) (A 2+B 2)-X0] 2+[(A 2Y -ABX -BC) (A 2+B 2)-Y0] 2=[(A 2X -Aby -AC) (A 2+B 2)] 2

Knowing that the coordinates of the point are (x0,y0) and the expression of the line is ax+by+c=0, then the distance from the point to the line is ((a*x0 + b*y0 + c) a 2+b 2) ).Absolute

Proof Method:

Definition forensics: By definition, the distance from the point p(x,y) to the line l:ax+by+c=0 is the perpendicular line from the point p to the line l.

The length of the segment should be set to the point p and the perpendicular line of the line is l', the vertical foot is q, then l'The slope of is b a then l'The analytic formula is y-y (b a) (x-x) and puts l and l'Synopid L and L'The coordinates of the intersection point q are ((b 2x aby ac) (a 2+b 2), a 2y abx bc) (a 2+b 2)) by the formula for the distance between the two points.

Get pq 2=[(b 2x -aby -ac) (a 2+b 2)-x0] 2

a^2y₀-abx₀-bc)/(a^2+b^2)-y0]^2

-a^2x₀-aby₀-ac)/(a^2+b^2)]^2

-abx₀-b^2y₀-bc)/(a^2+b^2)]^2

a(-by₀-c-ax₀)/a^2+b^2)]^2

b(-ax₀-c-by₀)/a^2+b^2)]^2

a^2(ax₀+by₀+c)^2/(a^2+b^2)^2

b^2(ax₀+by₀+c)^2/(a^2+b^2)^2

a^2+b^2)(ax₀+by₀+c)^2/(a^2+b^2)^2

ax₀+by₀+c)^2/(a^2+b^2)

So pq=|ax+by+c|a 2 + b 2), the formula is proven.

The derivation process of the distance formula from a point to a straight line: the distance formula d=(|) for ax+by+c=0ax_0+by_0+c|)/a~2+b~3)~(1/2)。The distance from the point to the straight line is the point where the target line is crossedPerpendiculars, from this point to the distance from the foot down.

Formula Description: The equation of the straight line in the formula.

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.

Point-to-straight distance definition:

The length from a point outside the line to the perpendicular segment of the line is called the distance from the point to the line. And the distance of this perpendicular segment is the shortest distance from any point to a straight line. Straight line ax+by+c=0 coordinates (xo,yo) then the distance from this point to this straight line is:

axo+byo+c│/√a+b）。

Of all the segments connected by a point outside the line and points on the line, the perpendicular segment is the shortest. The distance from a point to a straight line is called a perpendicular segment.

Process & Methodology:

1) Through the derivation of the distance formula from point to straight line, students can improve their understanding of the combination of numbers and shapes, and deepen their awareness of using "calculation" to deal with "figures".

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

The formula between the point and the distance between the point and the line is |ab|=[x2-x1) 2+(y2-y1) 2], the distance from the point to the straight line, that is, the perpendicular line of the target straight line through this point, the distance from this point to the perpendicular foot.

Through the derivation of the distance formula from point to straight line, students can improve their understanding of the combination of numbers and shapes, and deepen their awareness of using "calculation" to deal with "figures". Converts the distance relationship between two parallel straight lines to a point-to-straight distance.

The formula for the distance between a point and a line is:

Suppose the point p is (x0,y0), the equation for the line l is ax by c 0, and the distance between the point and the line d |ax0＋by0＋c|Remainder (a b).

Process & Method Objectives:

1) Through the derivation of the distance formula from the point to the straight line, students can improve their understanding of the combination of numbers and shapes, and deepen their awareness of using "calculation" to deal with "shapes".

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

Line to Line Distance Formula :

1. When two straight lines are parallel:

l1：ax＋by＋c＝0

l2：ax＋by＋d＝0

Distance |c-d|/√a＾2＋b＾2）

2. When two straight lines are not parallel: the distance is 0

The distance from the point to the line is frank.

Linear equation: ax by c 0

The coordinates of the point (x0, y0).

Then the distance formula from the point to the line: |ax0＋by0＋c|/√a＾2＋b＾2）<>

1. The formula for calculating the distance from the point to the line: let the equation of the straight 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: consider the point (x0, y0, z0) and the space line x-x1 l=y-y1 m=z-z1 n, and the liter has d=|(x1-x0,y1-y0,z1-z0)×(l,m,n)|/l2+m2+n2)。

2. The distance from the point to the bridge to the old line, that is, the perpendicular line of the straight line of the target through this point.

, from this point to the distance from the foot down.

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 point to straight line, students can improve their understanding of the combination of numbers and shapes, and deepen their awareness of using "calculation" to deal with "figures".

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

Linear equation: ax by c 0

The coordinates of the point (x0, y0).

Then the distance formula from the point to the line: |

ax0+by0+c|/√a^2+b^2)