The slope relationship of two straight lines symmetrical to any straight line

**Possibility one*** three lines intersect.

The purpose of this question is to let us illustrate.

How to use the tangent of 2 angles to indicate that these 2 angles are equal.

Let the angles between the three straight lines and the x-axis be angle 1, angle 2, and angle 3

The slope of the 3 straight lines is represented by r1, r2, r3 (write k1 b1, increase the length of the formula, and affect understanding).

The relationship given in the title is:

tan(angle1-angle3)=tan(angle3-angle2)According to tan( -=(tan -tan) (1+tan ·tan) substitution has (r1-r3) (1+r1*r3)=(r3-r2) (1+r3*r2) *The answer is in this***

ps: I guess maybe your title expression is a little unclear, if your question is the slope of k, just replace the r in my formula with k.

Possibility two*** three lines parallel.

r1=r2=r3

Line 1 and Line 2 are symmetrical with respect to Line 3, with the same angular difference.

arctan(k3,b3)-arctan(k2,b2)=arctan(k2,b2)-arctan(k1,b1)

arctan ((k3,b3)+(k2,b2))/(1- (k3,b3)*(k2,b2)))

arctan ((k2,b2)+(k1,b1))/(1- (k2,b2)*(k1,b1)))

k3,b3)+(k2,b2))/(1- (k3,b3)*(k2,b2)))

(k2,b2)+(k1,b1))/(1- (k2,b2)*(k1,b1)))

k3,b3)= (k1,b1) (k2,b2)^2 + 1)

With respect to the slopes of two straight lines symmetrical on the y-axis, they are inverse numbers to each other.

k1=-k2）

The proof is as follows:Let the slope of the two straight lines be k1 and k2, and the inclination angle is a and b.

If two straight lines are perpendicular, then the angle between them is 90 degrees.

So tan(a-b)=tan90=(tana-tanb) (1+tanatanb)=infinity.

Because tana=k1, tanb=k2.

So 1+tanatanb=1+k1k2=0.

Hence k1k2=-1.

Method 2:Let the slope of one line be tana and the other be tanb, and the angle between the two lines is b-a.

tan(b-a)=[tanb-tana]/[1+tana tanb]。

If 1 + tana tanb = 0, i.e. tana tanb = 1.

Then b - a = 90 degrees.

So, the conclusion is that if two straight lines are perpendicular to each other, then the product of the slopes of the two straight lines is -1.

Two straight lines are perpendicular and their slopes are reciprocal to each other. There are three types of positional relationships between two lines in a plane: coincident, parallel, and intersecting (perpendicular).

The slope is used to measure the slope of a slope. Mathematically, the slope of a straight line is equal everywhere and is a measure of how inclined a straight line is. Through algebra and geometry, the slope of a straight line can be calculated; The tangent slope of a point on the curve reflects how quickly the curve's variables change at that point.

Using calculus, you can calculate the tangent slope of any point in the curve. The concept of the slope of a straight line is equivalent to the slope in civil engineering and geography.

In the presence of a slope, the product of the slopes of the two straight lines is equal to -1 to prove that the oblique angle of the two straight lines is an acute angle and an obtuse angle (there is no slope at right angles).

Let the acute angle be , and the slope of the straight line is k; The obtuse angle is , and the slope of the straight line is k because the two lines are perpendicular, then =90°

k =tan =tan +90°= cot so k k=-cot tan =1

1. The relationship between the opposite numbers is inverse.

2. Let the slope of the straight line be k, and the slope of the two symmetrical straight lines is a and b, then there is such a relationship: (k-a) (1+ka)=(b-k) (1+kb) or assume that the inclination angle of the straight line is x, and half of the inclination angle of the two symmetrical oblique lines is x. In this way, the relationship can be obtained by using the tangent formula of the sum of the two angles.

3. The tangent value of the angle between a straight line and the direction of the transverse axis and the positive half axis of a plane Cartesian coordinate system is the slope of the straight line relative to the coordinate system. If the line is perpendicular to the x-axis, the tangent of the right angle is infinite, so the line has no slope.

4. When the slope of the straight line l exists, for the primary function y=kx+b (oblique truncation), k is the slope of the image (straight line) of the function.

Two straight lines are perpendicular, and the product of their slopes is -1 under the premise that both slopes exist; If there is no slope in one line, then the slope of the other line is 0. For two straight lines perpendicular to each other, their slopes are reciprocal of each other, so the product of their slopes is -1.

Slope calculation: ax+by+c=0, k=-a b, the slope formula of the straight line: k=(y2-y1) (x2-x1), the product of the slope of two perpendicular intersecting lines is -1:

k1*k2=-1, when k>0, the larger the angle between the straight line and the x-axis, the greater the slope; When k<0, the smaller the angle between the line and the x-axis, the smaller the slope.

Related formulas. When the slope of the straight line l exists, the oblique truncated y=kx+b. When x=0, y=b.

When the slope of the line l exists, the point slope y -y = k(x -x).

For any point on any function, its slope is equal to the tangent of its tangent at the angle of the positive direction of the x-axis, i.e., k=tan.

Slope calculation: straight line ax+by+c=0, slope k=-a b.

Let the line y=kx+b(k≠0), then there is:

The product of the slopes of two perpendicular intersecting lines is -1: k k =-1;

The slopes of two parallel lines are equal: k = k, and b ≠b

The proof is as follows:

Let the slope of the two straight lines be k1 and k2, and the inclination angle is a and b.

If two straight lines are perpendicular, then the angle between them is 90 degrees.

So tan(a-b)=tan90=(tana-tanb) (1+tanatanb)=infinity.

Because tana=k1, tanb=k2.

So 1+tanatanb=1+k1k2=0.

Hence k1k2=-1.

Introduction

The slope, also known as the "angular coefficient", is the tangent of the positive angle of a straight line to the abscissa axis, reflecting the inclination of the straight line to the horizontal plane, and the tangent value of the angle formed by a straight line and the positive semi-axis of the abscissa axis of a plane rectangular coordinate system is the slope of the straight line relative to the coordinate system.

If the line is perpendicular to the x-axis, the tangent of the right angle is infinite, so there is no slope of the line, and when the slope of the line l exists, for the primary function y=kx+b, (oblique truncation) k is the slope of the image of the function.

When the slope of the straight line l exists, the oblique truncated y=kx+b, when x=0, y=b, when the slope of the straight line l exists, the point oblique y1-y2=k(x1-x2), for any point on any function, its slope is equal to the tangent of the tangent to the positive direction of the x-axis, i.e., k=tan, the slope is calculated: ax+by+c=0, k=-a b.

1.If the x-axis is symmetrical, y'=-y is substituted into the original linear analytic formula, and the new analytic formula obtained is compared with the original one. If the y-axis is symmetrical, x'=-x can be substituted in the same way to compare.

In addition, through the definition of slope, the more intuitive method is that through the relationship between the tilt angles of the two symmetrical lines, it can be easily seen that the slopes of the two lines symmetrical on the x-axis or y-axis are all different by a negative sign, that is, the relationship between the opposite numbers.