How do you prove that two lines are perpendicular? The two line perpendicular formula how to prove

A: There are many ways. 1. The most basic method is to prove that the angle formed by the intersection of two lines is a right angle;

2. The inverse theorem using the Pythagorean theorem.

It is proved that in a triangle, it is enough to calculate that the square of the opposite side of a corner is equal to the sum of the squares of the other two sides;

3. Use isosceles triangles.

three lines in one" to prove that if one of the two lines is the base edge of the isosceles triangle, and the other line is the bisector of the top angle of the isosceles triangle or the middle line or height on the bottom edge, then the secondary two lines are perpendicular to each other;

4. Use right triangles.

From the sum theorem of the inner angles of the triangle, the sum of the two acute angles of a right triangle is equal to 90°, so the triangle with two acute angles must be a right triangle;

5. Use the diagonal of the diamond.

perpendicular to each other, if it can be proved that the two lines are diagonals of a diamond, they are perpendicular to each other;

6. Use the circumferential angle.

Corollary of the theorem: prove that the angle between two straight lines is the circumferential angle of the diameter of the circle, then it must be a right angle; 7. Using the relationship between the side lengths of triangles, as long as it is proved that the length of one side of a triangle is equal to half of the other side, then the triangle must be a right-angled triangle containing 30°.

8. Vector method, the product of two vectors = 0;

9. Analytical method, the product of the slope of the two lines = -1.

1. When two straight lines are perpendicular and the slope exists, the product of the slope is -1, that is, k1 k2=-1.

The general formula is a1a2+b1b2=0

2. Proof of the vertical formula of the general formula of two straight lines:

Let the line l1: a1x+b1y+c1=0 and the line l2: a2x+b2y+c2=0

necessity) l1 l2 k1 k2=-1

k1=-b1/a1, k2=-b2/a2

-b1/a1)(b2/a2)=-1 ∴（b1b2）/(a1a2)=-1

b1b2=-a1a2 ∴a1a2+b1b2=0

Sufficiency) a1a2+b1b2=0 b1b2=-a1a2

b1b2）(1/a1a2)=-1 ∴(b1/a1)(b2/a2)=-1

-b1/a1)(-b2/a2)=-1 ∵k1=-b1/a1, k2=-b2/a2

k1×k2=-1∴l1⊥l2

Proof that the two lines are parallel as follows:

The parallel formula is:

a2b1=a1b2, i.e.: a1b2-a2b1=0.

When two straight lines are perpendicular: k1k2=-1, then:

a1/b1=-b2/a2

a1a2 + b1b2 = 0 (in the presence of k).

The axiom of parallel lines is an important concept in geometry. The axiom of parallelism in Euclidean geometry can be expressed equivalently as "the only straight line parallel to the known straight line at a point of clear disturbance resistance".

The negative form of "a straight line that is not parallel to a known straight line at a point outside the straight line" or "at least two straight lines parallel to the known straight line at a point outside the straight line" can be used as an alternative to the axiom of parallelism in Euclidean geometry and deduce non-Euclidean geometry independent of Euclidean geometry.

If both lines are parallel to the third line, then the two lines are also parallel to each other. If a b, b c, then a c.

Extended Information: Determination of Parallel Lines.

1. The isotope angle is equal to the mu grind, and the two straight lines are parallel.

2. The internal staggered angles are equal, and the two straight lines are parallel.

3. The inner angle of the same side is complementary, and the two straight lines are parallel.

4. When two straight lines are parallel to the third straight line, the two straight lines are parallel.

5. In the same plane, two straight lines perpendicular to the same straight line are parallel to each other.

6. In the same plane, two straight lines parallel to the same straight line are parallel to each other.

7. Two straight lines that never intersect in the same plane are parallel to each other.

Parallel axioms of parallel lines.

1. After a point outside the straight line, there is only one straight line parallel to the known straight line.

2. The two parallel lines are truncated by the third straight line, the isotopic angle is equal, the internal error angle is equal, and the side internal angle is complementary.

Note: Only if the two parallel lines are truncated by the third answer, the isotopic angles will be equal, and the internal misalignment angles will be equal to the side internal angles.

Summary. 1. If any one of the two perpendicular lines is parallel, then the other perpendicular line is also perpendicular to this straight line;

2. The angle between two straight lines is equal to 90°, then the two straight lines are perpendicular (when the two straight lines intersect in the same plane, and the two straight lines are different in the same plane);

3. The straight lines in the vertical plane are also perpendicular to all the straight lines in this plane;

4. The tangent of the element is perpendicular to the straight line of the center of the circle and the tangent point;

5. The vertical bottom edge of the midline on the bottom edge of the isosceles triangle, and the vertical bottom edge of the bisector of the top angle of the isosceles triangle;

How can I tell if two straight lines are perpendicular or not?

1. If any one of the two perpendicular lines is parallel, then the other perpendicular line is also perpendicular to this straight line; 2. The angle formed by the two straight lines is equal to 90°, then the two straight lines are perpendicular (when the two straight lines intersect in the same plane, and the two states are different in the same plane when they are not in the same plane of Fanling); 3. The straight lines in the vertical plane are also perpendicular to all the straight lines in this plane; 4. The tangent of the element is perpendicular to the straight line of the center of the circle and the tangent point; 5. The vertical bottom edge of the midline on the bottom edge of the isosceles triangle, and the vertical bottom edge of the bisector of the top angle of the isosceles triangle;

Hope it helps.

The easiest way to do this is to have energy at right angles.

Visual experience will not be told. Introduce a simple measurement of pure fiber: use an ordinary tape measure, measure the two right-angled sides of the Qidan angle are 90cm, 120cm, make a mark, and then measure the straight-line distance of these two marks should be 150cm, which is 90 quietly.

This is the Pythagorean law principle, it is better to pull it with a tape measure of 5m, and its size should be m.

Inverse theorem proof using the Pythagorean theorem.

The inverse theorem of the Pythagorean theorem provides a computational method to prove that two lines are perpendicular, that is, to prove that one of the angles of a triangular annihilation vertical shape is equal to , because of the algebraic method, as long as the sum of the squares of the opposite sides of the right angle to be proved by the branch oak can be calculated to be equal to the sum of the squares of the other two sides.

For example, it is known that , and are two right-angled sides and hypotenuse sides of a straight triangle, which are the height of the hypotenuse, and it is verified that the triangle with , is a right-angled triangle.

Analysis: First of all, the maximum is determined by the method of measurement, and the triangle with , is a right-angled triangle, and only needs to be proved , and only needs to be proved , that is, because , so the problem is proved. (Testimonial).

Fourth, the use of "three-in-one" proof.

To prove that the two lines are perpendicular, if it can be proved that one of the two lines is the bottom edge of the isosceles triangle, and the other line is the bisector of the top angle of the isosceles triangle or the midline of the abrupt change on the bottom edge, then the two lines are perpendicular to each other.

Proof: Extend the intersection point, in and .

i.e

If two straight lines are perpendicular, the product of their slopes is -1

Perpendicular means that one line is at right angles to another, and these two straight lines are perpendicular to each other. It is usually indicated by the symbol " ".

Vertical nature:

In the same plane, there is one and only one line perpendicular to a known line. Vertically there will definitely be 90°.

Of all the segments connecting a point outside the line to the points on the line, the perpendicular segment is the shortest.

To put it simply: the perpendicular segment is the shortest.

Distance from point to line: The length from a point outside the line to the perpendicular segment of the line, called the distance from the point to the line.

Line and surface perpendicular: If a line is perpendicular to any line in a plane, the line is said to be perpendicular to each other in this plane.

The nature of the perpendicular line surface:

If a line is perpendicular to a plane, the line is perpendicular to all lines in the plane.

Passing through a point in space, there is only one and only one straight line perpendicular to the known plane.

If one of the two parallel lines is perpendicular to one plane, then the other line is also perpendicular to that plane.

Two straight lines perpendicular to the same plane are parallel.

1. When two straight lines are perpendicular and the slope exists, the product of the slope is -1, that is, k1 k2=-1.

The general formula is a1a2+b1b2=0

2. Proof of the vertical formula of the general formula of two straight lines:

The straight line L1: A1X+B1Y+C1=0, and the straight line L2: A2X+B2Y+C2=0

necessity) l1 l2 k1 k2=-1

k1=-b1/a1, k2=-b2/a2

b1 a1)(b2 a2)=-1 (b1b2) (a1a2)=-1

b1b2=-a1a2 ∴a1a2+b1b2=0

Sufficiency) a1a2+b1b2=0 b1b2=-a1a2

b1b2）(1/a1a2)=-1 ∴(b1/a1)(b2/a2)=-1

b1/a1)(-b2/a2)=-1 ∵k1=-b1/a1, k2=-b2/a2

k1×k2=-1∴l1⊥l2