In this lesson, we will take a closer look at the concept of the perpendicular to a line and prove an important theorem.
Let us first recall the definition of perpendicular lines. Next, we formulate and prove a theorem on two straight lines perpendicular to the third. Next, we give the definition of a perpendicular to a line, formulate and prove an important theorem that from any arbitrary point one can draw a single perpendicular to a given line.
In the end, we will solve several problems on the topic covered.
To begin with, let's remember an important fact: two intersecting straight lines are called perpendicular if they form four right angles.
Figure: 1. Perpendicular straight lines
AC⊥BD because the four corners are 90 °. We also recall that when any straight lines intersect, four angles are formed: 2 vertical angles, which are equal to each other, and a couple of equal vertical angles. a and b are adjacent corners. And by the adjacent angle theorem a + b \u003d 180 °.
Figure: 2. Intersection of straight lines
In the only case a \u003d b \u003d 90 °. In this case, straight lines AC and BD are called perpendicular.
Theorem 1: Two lines perpendicular to the third do not intersect.
Figure: 3. Drawing to Theorem 1
It follows that AA 1 and BB 1 have no points in common. Lines AA 1 and BB 1 can be extended infinitely, but they will not intersect. This is the meaning of the theorem.
Definition: Let lines AH and a be perpendicular. We know that for all four angles of these lines to be 90 ° each, it is necessary that one of them be right. Line segment AH is called a perpendicular drawn from point A to line a if lines AH and a perpendicular... In this case, point H is called the base of the perpendicular.
Figure: 4. Drawing to the definition of the perpendicular
In this case, the perpendicular is a line segment. This means that the perpendicular to the line is a segment.
Theorem 2: From a point not lying on a line, you can draw a perpendicular to this line, and moreover, only one.
Figure: 5. Drawing for Theorem 2
There are many points that do not lie on the line a. From any point A that does not lie on a given line, you can draw a perpendicular to the line. Moreover, this perpendicular is the only one.
Given: point A does not belong to line a.
Prove: there is only one segment AH, where AH.
Evidence:
1. Draw 2 equal angles. ∠ABS \u003d ∠МВС or ∠1 \u003d ∠2.
2. Equal corners can be overlaid. In this case, point A will go to point A 1. BA \u003d BA 1 (bend in a straight line BC).
3. Connect points A and A 1. We obtain point H. Angles ∠ВНА \u003d ∠3, ∠ВНA 1 \u003d ∠4.
4.
Therefore, triangles ВНА \u003d ВНA 1 according to the first sign of equality of triangles, that is, according to the angle and two adjacent sides. Equality of triangles implies equality of all elements. This means that ∠3 \u003d ∠4. These angles are opposite equal sides. Two adjacent ones are equal only if each of them is equal to 90 °. So, AH ^ BC. We have proved that from point A it is possible to draw a perpendicular to the line a.
Figure: 6. Drawing for the proof of Theorem 2 (1)
The uniqueness of the perpendicular drawn from the point A to the straight line will be proved by contradiction.
5. Suppose that two different perpendiculars can be drawn from point A to line a.
AH ⊥ a, AH 1 ⊥ a.
Figure: 7. Drawing for the proof of the uniqueness of the perpendicular
This is impossible, since 2 perpendiculars are drawn from different points of the line a, which have a common point A. We have received a contradiction, which means that our assumption is incorrect. Only one perpendicular can be drawn from point A to line a.
Example 1: Points A and C are on the same side of line a. The perpendiculars AB and CD to line a are equal.
1. Prove that ABD \u003d ∠CDB.
2. Find ∠ABS if ∠ADB \u003d 44 °.
Given: A) AB⊥ a, CD ⊥ a.
Prove: ∠ADB \u003d ∠CDB.
Evidence:
Figure: 8. Drawing for example 1 (a)
The proof is based on the concept of a perpendicular from a point to a line. Hence it follows that ADB \u003d CDB, as required.
Given: B) AB⊥ a, CD⊥ a. AB \u003d CD, ∠ADB \u003d 44 °. Find ∠ABS.
Evidence:
Let's execute an explanatory drawing:
Figure: 9. Drawing for example 1 (b)
1. ∆ABD \u003d ∆CDB. (AB \u003d CD, BD - general, ∠ABD \u003d ∠CDB). Equality of triangles implies equality of their corresponding elements. AD \u003d CB.
2. ∠ADB \u003d ∠CBD \u003d 44 °. Since these angles are opposite to equal sides AB and CD, respectively.
3.∠АВС \u003d 90 ° - 44 ° \u003d 46 °
Answer: 46 °.
In today's lesson, we examined the concept of a perpendicular to a line and proved a theorem about this perpendicular. In the next lesson, we will get acquainted with the median, bisector, and height of a triangle.
1. Alexandrov A.D., Verner A.L., Ryzhik V.I. and others. Geometry 7. - M .: Education.
2. Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. and others. Geometry 7. 5th ed. - M .: Education.
3. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolov, ed. Sadovnichy V.A. - M .: Education, 2010.
- Generalizing lesson in geometry in the 7th grade ().
- Straight line, segment ().
1. No. 13 (b). Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolov, ed. Sadovnichy V.A. - M .: Education, 2010.
2. One of the adjacent corners is 3 times the size of the other. Find these corners.
3. Lines BH and AH are mutually perpendicular and ∠BHM \u003d ∠AHC. Prove that НМ⊥НС.
4. No. 14 (d). Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolov, ed. Sadovnichy V.A. - M .: Education, 2010.
Theorem. From a point not lying on a straight line, you can draw a perpendicular to this straight line.
Evidence.Let A be a point not lying on a given line a (Fig. 56, a). Let us prove that from point A one can draw a perpendicular to the line a. Bend the plane mentally along the straight line a (Fig. 56, b) so that the half-plane with the boundary a, containing the point A, overlaps the other half-plane. In this case, point A will be superimposed on some point. Let us denote it by the letter B. We unbend the plane and draw a straight line through the points A and B.
Let H be the point of intersection of lines AB and a (Fig. 56, c). When the plane bends again along a straight line, point H will remain in place. Therefore, the HA ray will overlap with the HB ray, and therefore the angle 1 will coincide with the angle 2. Thus, ∠1 \u003d ∠2. Since angles 1 and 2 are adjacent, their sum is 180 °, so each of them is straight. Therefore, the segment AH is the perpendicular to the line a. The theorem is proved.
26. Prove a theorem on the uniqueness of the perpendicular to the line. (Fig. 57 in the tutorial)
Theorem. From a point not lying on a straight line, it is impossible to draw two perpendiculars to this straight line.
Evidence.Let A be a point not lying on a given line a (see Fig. 56, a). Let us prove that it is impossible to draw two perpendiculars from the point A to the line a. Suppose that from point A you can draw two perpendiculars AH and AK to the straight line a (Fig. 57). We mentally bend the plane along the straight line a so that the half-plane with the boundary a, containing the point A, is superimposed on the other half-plane. When bending, points H and K remain in place, point A is superimposed on some point. Let us denote it by the letter B. In this case, the segments AH and AK are superimposed on the segments BH and BK.
Angles AHB and AKB are unfolded, since each of them is equal to the sum of two right angles. Therefore, points A, H and B lie on one straight line and also points A, K and B lie on one straight line.
Thus, we got that two lines AH and AK pass through points A and B. But this cannot be. Consequently, our assumption is incorrect, which means that two perpendiculars cannot be drawn from point A to the line a. The theorem is proved.
http://mthm.ru/geometry7/perpendicular
Perpendicular straight lines appear in almost every geometric problem. Sometimes the perpendicularity of the lines is known from the condition, and in other cases the perpendicularity of the lines has to be proved. To prove the perpendicularity of two straight lines, it is enough to show, using any geometric methods, that the angle between the straight lines is equal to ninety degrees.
And how to answer the question "are the lines perpendicular" if the equations that define these lines on a plane or in three-dimensional space are known?
To do this, use necessary and sufficient condition for the perpendicularity of two straight lines... Let us formulate it as a theorem.
Theorem.
a and b it is necessary and sufficient that the directing vector of the straight line a was perpendicular to the direction vector of the straight line b.
The proof of this condition of perpendicularity of straight lines is based on the definition of the direction vector of the line and on the definition of perpendicular lines.
Let's add specifics.
Let a rectangular Cartesian coordinate system be introduced on the plane Oxy and equations of a straight line on a plane of some form are given, which determine the straight lines a and b... We denote the direction vectors of straight lines and and b as well as respectively. Equations of lines a and bit is possible to determine the coordinates of the direction vectors of these straight lines - we obtain and. Then, for the perpendicularity of the lines a and b it is necessary and sufficient that the condition of perpendicularity of vectors and is fulfilled, that is, that the scalar product of vectors and is equal to zero: 
.
So, a and b in a rectangular coordinate system Oxy on the plane has the form , where and are direction vectors of straight lines a and b respectively.
This condition is convenient to use when the coordinates of the direction vectors of straight lines are easily found, as well as when the straight line a and b correspond to canonical equations of a straight line on a plane or parametric equations of a straight line on a plane.
Example.
In a rectangular coordinate system Oxy three points are given. Are the straight lines perpendicular AB and AS?
Decision.
Vectors and are direction vectors of straight lines AB and AS... Referring to the article, the coordinates of the vector by the coordinates of the points of its beginning and end, we calculate 
... The vectors and are perpendicular, since 
... Thus, the necessary and sufficient condition for the perpendicularity of the lines AB and AS... Therefore, the straight lines ABand AS perpendicular.
Answer:
yes, straight lines are perpendicular.
Example.
Are straight and 
perpendicular?
Decision.
The directing vector of the straight line, and is the directing vector of the straight line ... Let's calculate the dot product of vectors and: 
... It is nonzero, therefore, the direction vectors of the lines are not perpendicular. That is, the condition of perpendicularity of lines is not met, therefore, the original lines are not perpendicular.
Answer:
no, straight lines are not perpendicular.
Similarly, a necessary and sufficient condition for the perpendicularity of straight lines a and b in a rectangular coordinate system Oxyz in three-dimensional space has the form 
where 
and 
- direction vectors of straight lines a and b respectively.
Example.
Are the rectangular lines perpendicular Oxyz in three-dimensional space by equations 
and?
Decision.
The numbers in the denominators of the canonical equations of the straight line in space are the corresponding coordinates of the directing vector of the straight line. And the coordinates of the directing vector of the straight line, which is given by the parametric equations of the straight line in space, are the coefficients at the parameter. In this way, 
and - direction vectors of given lines. Let's find out if they are perpendicular: 
... Since the dot product is zero, these vectors are perpendicular. This means that the condition of perpendicularity of the given lines is satisfied.
Answer:
straight lines are perpendicular.
To check the perpendicularity of two straight lines on a plane, there are other necessary and sufficient perpendicularity conditions.
Theorem.
For perpendicularity of straight lines a and b on the plane it is necessary and sufficient that the normal vector of the line a was perpendicular to the normal line vector b.
The voiced condition of perpendicularity of straight lines is convenient to use if the coordinates of the normal vectors of straight lines can be easily found from the given equations of straight lines. This statement corresponds to the general equation of the line of the form 
, the equation of a straight line in segments and the equation of a straight line with a slope.
Example.
Make sure straight 
and are perpendicular.
Decision.
Given the equations of the straight lines, it is easy to find the coordinates of the normal vectors of these straight lines. Is the normal vector of the line ... We rewrite the equation as 
, whence the coordinates of the normal vector of this line are visible:.
The vectors and are perpendicular, since their dot product is zero: 
... Thus, the necessary and sufficient condition for the perpendicularity of the given lines is fulfilled, that is, they are really perpendicular.
In particular, if the straight line a on the plane defines the equation of a straight line with a slope of the form, and a straight line b - form, then the normal vectors of these lines have coordinates and, respectively, and the condition of perpendicularity of these lines is reduced to the following relation between the slope coefficients.
Example.
Are the straight lines and perpendicular?
Decision.
The slope of the line is equal and the slope of the line is. The product of the slopes is equal to minus one, therefore, the straight lines are perpendicular.
Answer:
given straight lines are perpendicular.
One more condition of perpendicularity of straight lines on a plane can be voiced.
Theorem.
For perpendicularity of straight lines a and b on the plane, it is necessary and sufficient that the direction vector of one straight line and the normal vector of the second straight line are collinear.
Obviously, it is convenient to use this condition when the coordinates of the directing vector of one straight line and the coordinates of the normal vector of the second straight line are easily found, that is, when one straight line is given by the canonical equation or parametric equations of a straight line on a plane, and the second is either the general equation of a straight line, or the equation of a straight line in segments, or the equation of a straight line with a slope.
Example.
Are straight and perpendicular?
Decision.
Obviously, is the normal vector of the straight line, and is the directing vector of the straight line. The vectors and are not collinear, since the condition of collinearity of two vectors does not hold for them (there is no such real number t, at which). Therefore, the given lines are not perpendicular.
Answer:
straight lines are not perpendicular.
21. Distance from point to straight line.
The distance from a point to a straight line is defined as the distance from a point to a point. Let's show how this is done.
Let a straight line be given on a plane or in three-dimensional space a and point M 1not lying on a straight line a... Let's draw through the point M 1 straight bperpendicular to the straight line a... We denote the point of intersection of the lines a and b as H 1... Line segment M 1 H 1 called perpendiculardrawn from point M 1 to straight a.
Definition.
Distance from point M 1 to straight a called the distance between points M 1 and H 1.
However, it is more common to determine the distance from a point to a straight line, in which the length of the perpendicular appears.
Definition.
Distance from point to line Is the length of the perpendicular drawn from a given point to a given straight line.
This definition is equivalent to the first definition of the distance from a point to a line.
Note that the distance from a point to a line is the smallest of the distances from that point to points on a given line. Let's show it.
Let's take on a straight line a point Qnot coinciding with point M 1... Line segment M 1 Q called obliquedrawn from point M 1 to straight a... We need to show that the perpendicular drawn from the point M 1 to straight aless than any oblique drawn from a point M 1 to straight a... It really is: triangle M 1 QH 1 rectangular with hypotenuse M 1 Q, and the length of the hypotenuse is always greater than the length of any of the legs, therefore, 
.
22. Plane in the space R3. Equation of the plane.
A plane in a Cartesian rectangular coordinate system can be given by the equation, 
which is called general equationplane.
Definition.The vector is perpendicular to the plane and is called it normal vector.
If in a rectangular coordinate system the coordinates of three points that do not lie on one straight line are known, then the plane equation is written in the form: 
.
Having calculated this determinant, we obtain the general equation of the plane.
Example.Write the equation of the plane passing through the points.
Decision:
Plane equation:.
23. Study of the general equation of the plane.
DEFINITION 2. Any vector perpendicular to a plane is called the normal vector of this plane.
If a fixed point is known M 0 (x 0 , y 0 , z 0) lying in a given plane and a vector perpendicular to this plane, then the equation of the plane passing through the point M 0 (x 0 , y 0 , z 0), perpendicular to the vector, has the form
A(x-x 0)+ B(y-y 0) + C(z-z 0)= 0. (3.22)
Let us show that equation (3.22) is a general equation of the plane (3.21). To do this, we expand the brackets and put the free term in brackets:
.Ax + By + Cz +(-Ax 0 - By -Cz 0)= 0
By designating D = -Ax 0 - By -Cz 0, we obtain the equation Ax + By + Cz + D= 0.
Objective 1. Make the equation of the plane passing through point A, perpendicular to the vector, if A(4, -3, 1), B(1, 2, 3).
Decision. Find the normal vector of the plane:
To find the equation of the plane, we use equation (3.22):
Answer: -3x + 5y + 2z + 25 = 0.
Objective 2. Equate a plane through a point M 0 (-1, 2, -1), perpendicular to the axis OZ.
Decision. As a normal vector of the desired plane, you can take any vector lying on the OZ axis, for example, then the plane equation
Answer: z + 1 = 0.
24. Distance from point to plane.
The distance from a point to a plane is determined through the distance from a point to a point, one of which is a given point, and the other is the projection of a given point onto a given plane.
Let a point be given in three-dimensional space M 1 and plane. Let's draw through the point M 1straight aperpendicular to the plane. Let us denote the point of intersection of the straight line a and planes like H 1... Line segment M 1 H 1 called perpendiculardropped from the point M 1 on a plane, and point H 1 – base of perpendicular.
Definition.
Is the distance from a given point to the base of a perpendicular drawn from a given point to a given plane.
The most common definition is the distance from a point to a plane in the following form.
Definition.
Distance from point to plane Is the length of the perpendicular dropped from a given point to a given plane.
It should be noted that the distance from the point M 1 to the plane, defined in this way, is the smallest of the distances from a given point M 1 to any point on the plane. Indeed, let the point H 2 lies in the plane and differs from the point H 1... Obviously the triangle M 2 H 1 H 2 is rectangular, in it M 1 H 1 - leg, and M 1 H 2 Is the hypotenuse, therefore 
... By the way, the segment M 1 H 2 called obliquedrawn from point M 1 to the plane. So, a perpendicular dropped from a given point to a given plane is always less than an inclined line drawn from the same point to a given plane.
If the straight line passes through two given points ,
then her the equationwritten in the form : 
.
Definition. The vector is called guiding a straight line vector if it is parallel or belongs to it.
Example.Write the equation of a straight line passing through two given points 
.
Solution: We use the general formula of a straight line passing through two given points: - the canonical equation of a straight line passing through points and. Vector - directing vector of a straight line.
26. Mutual arrangement of straight lines in space R3.
Let's move on to the options for the relative position of two straight lines in space.
Firstly, two lines can coincide, that is, have infinitely many common points (at least two common points).
Secondly, two straight lines in space can intersect, that is, have one common point. In this case, these two straight lines lie in a certain plane of three-dimensional space. If two straight lines in space intersect, then we come to the concept of an angle between intersecting straight lines.
Thirdly, two straight lines in space can be parallel. In this case, they lie in the same plane and have no common points. We recommend to study the article parallel lines, parallelism of lines.
After we have given the definition of parallel straight lines in space, it should be said that they are the directing vectors of a straight line because of their importance. Any nonzero vector lying on this straight line or on a straight line that is parallel to the given one will be called the directing vector of the straight line. The direction vector of a straight line is very often used in solving problems related to a straight line in space.
Finally, two straight lines in three-dimensional space can be crossed. Two lines in space are called intersecting if they do not lie in the same plane. Such a mutual arrangement of two straight lines in space leads us to the concept of the angle between crossing straight lines.
Of particular practical importance is the case when the angle between intersecting or crossing lines in three-dimensional space is equal to ninety degrees. Such straight lines are called perpendicular (see the article perpendicular lines, perpendicularity of lines).
27. Mutual arrangement of a straight line and a plane in the space R3.
A straight line can lie on a given plane, be parallel to a given plane or intersect it at one point, see the following figures.
If, then this means that. And this is possible only when the straight line lies on the plane or is parallel to it. If the straight line lies on a plane, then any point of the straight line is a point on the plane and the coordinates of any point on the straight line satisfy the equation of the plane. Therefore, it is enough to check whether a point lies on the plane. If, then point 
- lies on the plane, which means that the line itself lies on the plane.
If, a, then the point on the straight line does not lie on the plane, which means that the straight line is parallel to the plane.
The theorem is proved.
A straight line (a segment of a straight line) is designated by two capital letters of the Latin alphabet or one small letter. The point is indicated only by a capital Latin letter.
Lines may not intersect, intersect or coincide. Intersecting straight lines have only one common point, non-intersecting straight lines - no common point, coinciding straight lines have all points in common.
Definition. Two straight lines intersecting at right angles are called perpendicular. The perpendicularity of straight lines (or their segments) is indicated by the perpendicularity sign "⊥".
For instance:
Your AB and CD (fig. 1) intersect at the point ABOUT and ∠ AOC = ∠VOS = ∠AOD = ∠BOD \u003d 90 °, then AB ⊥ CD.
If AB ⊥ CD (fig. 2) and intersect at the point AT, then ∠ ABC = ∠ABD \u003d 90 °
Properties of perpendicular lines
1. Through point AND (Fig. 3) only one perpendicular line can be drawn AB to straight CD; the rest of the lines passing through the point AND and crossing CD, are called oblique straight lines (Fig. 3, straight lines AE and AF).
2. From point A you can lower the perpendicular to a straight line CD; perpendicular length (segment length AB) drawn from the point AND on a straight line CD, is the shortest distance from A before CD (fig. 3).
Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify a specific person or contact him.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you leave a request on the site, we may collect various information, including your name, phone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notifications and messages.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, competition or similar promotional event, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose information received from you to third parties.
Exceptions:
- If it is necessary - in accordance with the law, court order, in court proceedings, and / or on the basis of public requests or requests from government authorities on the territory of the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other socially important reasons.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to an appropriate third party - the legal successor.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.
Respect for your privacy at the company level
In order to make sure that your personal information is safe, we bring the rules of confidentiality and security to our employees, and strictly monitor the implementation of confidentiality measures.