1. Prove that a perpendicular drawn from a point to a straight line is less than any inclined slope drawn from the same point to this straight line. 2. Prove that all points of each of two parallel lines are equidistant from the other line. 3. Solve problem No. 274.
3.Indicate the inclined lines drawn from point A to line BD. 4. What is the distance from a point to a line called? 5. What is the distance between two parallel lines called? 1. Specify a segment that is a perpendicular drawn from point A to line BD. 2. Explain what segment is called an inclined segment drawn from a given point to a given line.
Find the distance from point A to straight line a. Given: KA = 7 cm. Find: the distance from point A to straight line a. Rice. 4.192.
1. Explain how to plot a segment equal to the given one on a given ray from its beginning. 2. Explain how to plot an angle equal to a given one from a given ray. 3. Explain how to construct the bisector of a given angle. 4. Explain how to construct a line passing through a given point lying on a given line and perpendicular to this line. 5. Explain how to construct the midpoint of a given segment. Constructing a triangle using three elements.
1 row. Given: Fig. 4.193. Construct: ABC such that AB = PQ, A = M, B = N, using a compass and a ruler without divisions. 2nd row. Given: Fig. 4.194. Construct: ABC such that AB = MN, AC = RS, A = Q, using a compass and a ruler without divisions. 3rd row. Given: Fig. 4.195. Construct: ABC such that AB = MN, BC = PQ, AC = RS, using a compass and a ruler without divisions.
D C Constructing a triangle using two sides and the angle between them. hk h Let's construct ray a. Let us set aside the segment AB equal to P 1 Q 1 . Let's construct an angle equal to this one. Let us set aside the segment AC equal to P 2 Q 2 . B A Δ ABC is the desired one. Given: Segments P 1 Q 1 and P 2 Q 2, Q 1 P 1 P 2 Q 2 a k Doc: By construction AB=P 1 Q 1, AC=P 2 Q 2, A= hk. Build. Construction.
For any given segments AB=P 1 Q 1, AC=P 2 Q 2 and a given undeveloped hk, the required triangle can be constructed. Since straight line a and point A on it can be chosen arbitrarily, there are infinitely many triangles that satisfy the conditions of the problem. All these triangles are equal to each other (according to the first sign of equality of triangles), therefore it is customary to say that this problem has a unique solution.
D C Constructing a triangle using a side and two adjacent angles. h 1 k 1 , h 2 k 2 h 2 Let's construct ray a. Let us set aside the segment AB equal to P 1 Q 1 . Let's construct an angle equal to the given h 1 k 1 . Let's construct an angle equal to h 2 k 2 . B A Δ ABC is the desired one. Given: Segment P 1 Q 1 Q 1 P 1 a k 2 h 1 k 1 N Doc: By construction AB = P 1 Q 1 , B = h 1 k 1 , A = h 2 k 2 . Construct Δ. Construction.
C Let's build a ray a. Let us set aside the segment AB equal to P 1 Q 1 . Let's construct an arc with a center at point A and radius P 2 Q 2 . Let's construct an arc with center at t.B and radius P 3 Q 3 . B A Δ ABC is the desired one. Given: Segments P 1 Q 1, P 2 Q 2, P 3 Q 3. Q 1 P 1 P 3 Q 2 a P 2 Q 3 Construction of a triangle using three sides. Doc: By construction AB=P 1 Q 1, AC=P 2 Q 2 CA= P 3 Q 3, i.e. the sides Δ ABC are equal to these segments. Construct Δ. Construction.
A problem does not always have a solution. In any triangle, the sum of any two sides is greater than the third side, therefore, if any of the given segments is greater than or equal to the sum of the other two, then it is impossible to construct a triangle whose sides would be equal to these segments.
Problem No. 286, 288.
Homework: § 23, 37 - repeat, § 38!!! Questions 19, 20 p. 90. Solve problems No. 273, 276, 287, Solve problem No. 284.
Geometry lesson in 7th grade
(using the technology of the system-activity approach)
Mathematics teacher at the Kitovskaya MSOSH, Shuisky district, Ivanovo region, Nadezhda Mikhailovna Korovkina.
- Lesson topic: “Construction problems.
- Construction of a triangle using three elements." (using presentation)
Stages of a lesson in mastering new knowledge.
1. Motivation (self-determination) for educational activities:
involves the student’s conscious entry into the space of learning activity.
For this purpose, the student’s motivation for learning activities in the lesson is organized, namely:
1) the requirements for it from educational activities are updated (“must”);
2) conditions are created for the emergence of an internal need for him to be included in educational activities (“I want”);
3) the thematic framework (“I can”) is established.
Assumes:
1) updating learned ways of doing things, sufficient for the construction of new knowledge, their generalization;
2) recording of individual difficulties by students in performing a trial educational action or justifying it.
3. Identifying the location and cause of the difficulty.
At this stage, students identify the location and cause of the difficulty.
To do this they must:
correlate your actions with the method of action used (algorithm, concept, etc.), and on this basis, identify and record in external speech the cause of the difficulty - those specific knowledge, skills or abilities that are lacking to solve the original problem and problems of this class or kind of in general.
Students determine the topic of the lesson and formulate their own goals.
Students communicatively think about a project for future educational activities:
choose a method
build a plan to achieve the goal;
determine means, resources and timing.
This process is led by the teacher: at first with the help of introductory dialogue, then with stimulating dialogue, and then with the help of research methods
6. Implementation of the constructed project. (“Discovery” of new knowledge)
At this stage, students put forward hypotheses and build models of the original problem situation. Various options proposed by students are discussed and the optimal option is selected, which is recorded in the language verbally and symbolically.
The constructed method of action is used to solve the original problem that caused the difficulty.
In conclusion, the general nature of the new knowledge is clarified and the overcoming of the previously encountered difficulty is recorded.
7. Primary consolidation with pronunciation in external speech.
Students, in the form of communicative interaction (frontally, in groups, in pairs), solve standard tasks for a new method of action, pronouncing the solution algorithm out loud.
Students independently perform tasks of a new type, self-test them, step by step comparing them with the standard, identify and correct possible errors, determine methods of action that cause them difficulties and they have to refine them.
The emotional focus of the stage is to organize a situation of success for each student, motivating him to engage in further cognitive activity.
9. Inclusion in the knowledge system and repetition.
At this stage, the boundaries of applicability of new knowledge are identified and tasks are performed in which a new method of action is provided as an intermediate step.
10. Reflection on learning activities in the lesson.
At this stage, new content learned in the lesson is recorded, and reflection and self-assessment of students’ own learning activities is organized.
11. Lesson summary.
At this stage, the goal of the educational activity and its results are correlated, the degree of their compliance is recorded, and further goals of the activity are outlined.
Advantages of a lesson using the system-activity method
Children learn better what they discovered themselves, and not what they received ready-made and memorized. Thus, such a lesson provides triple effect:
high-quality knowledge acquisition;
development of intelligence and creativity;
education of an active personality.
- Lesson topic: “Construction problems. Construction of a triangle using three elements."
Lesson objectives:
Educational: introducing students to problems of constructing triangles using three elements; convey the material being studied to students as much as possible;
Developmental: develop thinking, memory, and the ability to freely use a compass;
Educational: try to increase the activity and independence of students when performing practical tasks.
Equipment: school compass, ruler, interactive whiteboard, projector, laptop.
DURING THE CLASSES
1. Motivation for educational activities.
Remember: what type of tasks can be classified as shown on the slides?
(Tasks on constructing an angle equal to a given one and a task on constructing the bisector of an angle.)
2. Updating and recording individual difficulties in a trial action.
Teacher: Let's remember how to construct an angle equal to a given one, and how to construct a bisector of a given angle. (slides No. 1 -3) Frontal conversation.
3. Identifying the location and causes of the difficulty.
Teacher: What do you think we will talk about in class today? (about construction tasks)
Think about what we will build in accordance with the topic that we are going through. Slide No. 4. (Students’ answer: triangles)
Teacher: So, today we will learn to build triangles.
How many elements are enough to know for the triangles to be equal? (three) Let's remember what signs of equality of triangles do you know? (students' answers)
Therefore, a triangle equal to this one can also be constructed using three elements.
In construction problems we will use only a compass and a ruler.
4. Formulating the topic and purpose of the lesson.(slide 6)
Teacher: Try to formulate the topic and purpose of today's lesson.
(students' answers)
Lesson topic: “Constructing a triangle using three elements” (write it down in a notebook)
The purpose of the lesson: get acquainted with the tasks of constructing triangles using three elements.
Teacher: What tasks will we set for ourselves? (formulated by students)
1) Get acquainted with the tasks of constructing triangles using three elements.
2) Derive an algorithm for solving problems on constructing triangles.
3) Try to independently construct triangles using three elements.
5. Construction of a project for getting out of the difficulty.
Teacher: Any construction task includes four main stages:
analysis; construction; proof; study.
Analysis and research of the problem are as necessary as the construction itself. It is necessary to see in which cases the problem has a solution, and in which there is no solution.
Conducted orally analysis building tasks(we sort it out together with students). A project is being built that will need to be put into action.
6 .Implementation of the constructed project. (“Discovery” of new knowledge)
Group work. (slide 7)
Exercise: Construct a triangle using three elements. Derive an algorithm for constructing triangles.
Group 1 - construction of a triangle using two sides and the angle between them.
Group 2 - construction of a triangle using a side and two adjacent angles.
Group 3 - construction of a triangle on three sides.
7. Primary consolidation with pronunciation in external speech.
Group report. One of the students in the group speaks at the blackboard, all other students make appropriate notes in their notebooks. (slides No. 9-16)
1 group. Student answer.
Constructing a triangle using two sides and the angle between them. (slides No. 10-12)
Given: segments P 1 Q 1 and P 2 Q 2 angle hk;
Describes how to construct a triangle using two sides and the angle between them.
An algorithm for constructing a triangle using two sides and the angle between them is derived and written down in a notebook.
Construction algorithm
1. Let's draw a straight line A.
AB, equal to the segment P 1 Q 1 .
3. Construct an angle TO YOU, equal to the given angle hk .
4. On the beam AM put aside the segment AC, equal to the segment P 2 Q 2.
5. Let's draw a segment B.C. .
6. Constructed triangle ABC- sought after.
Physical education minute. (slides No. 19-22)
II group.
Student answer.
2 . Constructing a triangle using a side and its adjacent angles. (Slides No. 13-15)
Given: segment; 2 corners;
A student explains how to construct a triangle using a side and two adjacent angles. The algorithm for constructing a triangle is derived.
Construction algorithm
1. Let's draw a beam AK starting at a point A.
2. Let us plot the angle from the beginning of the ray using a compass WITH 1 AB, equal to the angle hk .
3. From the beginning of the ray we will set aside a segment AB, equal to the segment P 1 Q 1 .
4. Construct an angle ABC 2 , equal to the angle mn .
5. Point of intersection of rays AC 1 And Sun 2 denote by a dot WITH.
6. Constructed triangle ABC- sought after.
III group.
Student answer . Constructing a triangle using three sides. (slides No. 16-18)
Given “P 1 Q 1”, “P 2 Q 2”, “P 3 Q 3”. Required to construct ABC
A student talks about how to construct a triangle using three sides. The algorithm is displayed.
Construction algorithm
1
. Let's make a direct A.
2. Using a compass, draw a segment on it AB, equal to the segment R 1 Q 1 .
3. Construct a circle with center A and radius R 3 Q 3 .
4. Construct a circle with center IN and radius P2Q 2 .
5. Let us denote one of the points of intersection of these circles by a point WITH.
6. Let's draw segments AC And Sun.
7. Constructed triangle ABC- sought after.
8. Independent work with self-test according to the standard.(slides 23 -24)
Task (independently, followed by self-test)
Construct a triangle ODE if OD = 4 cm, DE = 2 cm, EO = 3 cm.
After constructing any triangle, independently prove that the resulting triangle is the one you are looking for, and, if possible, conduct research.
9. Homework: No. 290 p.38. (slide 25)
10. Summing up the lesson. (slide 26)
What goal did we set for ourselves at the beginning of the lesson?
Have we solved those problems? which ones have you set for yourself?
11. Reflection on learning activities in the lesson.(slide 27)
Got it
Still need to work
Didn't understand the material well.
Methodological materials used for the lesson:
Presentation for the lesson.
Presentation from the site “Ur ok Mathematics” Igor Zhaborovsky. (slide No. 24)
Textbook of geometry for grades 7-9, ed. Atanasyan L.S. Moscow "Enlightenment" 2008
View presentation content
"present.built.triug.7 class"
(System-activity teaching method)
Korovkina Nadezhda Mikhailovna – mathematics teacher at Kitovskaya Secondary School of Shuisky district
Construction tasks
Constructing an angle equal to a given one
Task
Given:
Construction:
Build:
6. okr(E,BC)
2. en(A,r) ; g-any
KOM = A
3. en(A; g) A= B; C
7. okr(E,BC) okr(O,g)= K;K 1
4. okr(O,g)
5. okr(O,g) OM= E
Task
Construct the bisector of a given angle
Given :
Build :
Beam AE - bisector A
Construction :
5. okr(B; g 1) okr(C; g 1)= E; E 1
1. env(A; r); g-any
6. E-inside A
2. en(A; g) A= B; C
3. en(V;r 1)
4. en(C;g 1)
8 . AE- searched
Group work
Constructing a triangle using three elements
- 1 group- construction of a triangle using two sides and the angle between them.
- 2nd group- construction of a triangle using two angles and the side between them.
- 3 group- construction of a triangle on three sides.
1. segments P 1 Q 1 and P 2 Q 2.
Construction
Construction algorithm
1. Let's draw a straight line A .
2. Put it on it using
compass segment AB, equal
segment P 1 Q 1 .
3. Construct an angle TO YOU,equal
this angle hk .
4. On the beam AM put aside the segment
AC, equal to the segment P 2 Q 2 .
5. Let's draw a segment B.C. .
6. Constructed triangle
ABC- sought after.
1. segments P 1 Q 1.
2. angle hk and mn
You need to: use a compass and a ruler without scale divisions to construct a triangle.
Construction algorithm
1. Let's draw a beam AK with the beginning
at the point A .
2. Let us postpone from the beginning of the ray from
using a compass angle WITH 1 AB ,
equal to angle hk .
3. From the beginning of the beam we will postpone
line segment AB, equal to the segment P 1 Q 1 .
4. Construct an angle ABC 2 , equal
corner mn .
5. Point of intersection of rays
AC 1 And Sun 2 denote by a dot WITH .
6. Constructed triangle
ABC- sought after.
Construction
Segments: P 1 Q 1, P 2 Q 1, P 1 Q 1
You need to: use a compass and a ruler without scale divisions to construct a triangle.
Construction algorithm
1. Let's draw a straight line A .
2. Put it on it using
compass segment AB, equal
segment R 1 Q 1 .
3. Construct a circle with
center A and radius R 3 Q 3 .
4. Construct a circle with
center IN and radius R 2 Q 2 .
5. One of the intersection points
denote these circles
dot WITH .
6. Let's draw segments AC And Sun .
7. Constructed triangle
ABC- sought after.
Construction
We quickly got up from our desks
And they walked on the spot
- And now we smiled
- Higher, higher we reached.
Straighten your shoulders
raise, lower,
Turn to the left, turn to the left.
And sit down at your desk again.
Task (on one's own)
Construct a triangle using its three sides
Construction algorithm
1. Let's draw a straight line A .
2. Using a compass, draw a segment on it OD= 4 cm
3. Construct a circle with
center ABOUT and radius OE = 2 cm.
4. Construct a circle with
center D and radius DE = 3 cm.
5. Let us denote one of the intersection points of these circles
dot E .
6. Let's draw segments OE And DE .
7. Constructed triangle
OED- sought after.
Given: OD = 4 cm,
DE = 3 cm,
EO = 2 cm.
Igor Zhaborovsky © 2011
UROKI MATHEMATICS .RU
- P. 38 p.84 (learn the algorithm)
- No. 291 (a,b)
The work contains 29 slides for the lesson on the topic "Constructing triangles using three elements"
n1) Get acquainted with the problems of constructing triangles;
n2) Derive an algorithm for solving problems on constructing triangles.
n3) Try to independently construct triangles using three elements.
Construction algorithm
1. Let's draw a straight line A.
2. Put it on it using
compass segment AB, equal
segment M 1 N1.
3. Construct an angle TO YOU, equal
this angle hk.
4. On the beam AM put aside the segment
AC, equal to the segment M 2 N2 .
5. Let's draw a segment B.C..
6. Constructed triangle
ABC- sought after.
Construction algorithm
1. Let's draw a beam AK with the beginning
at the point A.
2 From the beginning of the ray we will postpone
line segment AB, equal to the segment M 1N1.
3. Let us postpone from the beginning of the ray from
using a compass angle C1AB,
equal to angle hk.
4. Construct an angle ABC2, equal
corner mn.
5. Point of intersection of rays
AC1 And BC2 denote by a dot WITH.
6. Constructed triangle
ABC- sought after.
Construction algorithm
1. Let's draw a straight line A.
AB, equal to the segment M 1N1.
3. Construct a circle with
center A and radius M 2 N2 .
4. Construct a circle with
center IN radius M 3 N3 .
dot WITH.
6. Let's draw segments AC And Sun.
7. Constructed triangle ABC- sought after.
View document contents
“presentation for the geometry lesson “Constructing triangles”, grade 7”
Construction tasks
Constructing an angle equal to a given one
Task
Given:
Construction:
Build:
6. okr(E,BC)
2. en(A,r) ; g-any
KOM = A
3. en(A; g) A= B; C
7. okr(E,BC) okr(O,g)= K;K 1
4. okr(O,g)
5. okr(O,g) OM= E
Task
Construct the bisector of a given angle
Given :
Build :
Beam AE - bisector A
Construction :
5. okr(B; g 1) okr(C; g 1)= E; E 1
1. env(A; r); g-any
6. E-inside A
2. en(A; g) A= B; C
3. en(V;r 1)
4. en(C;g 1)
8 . AE- searched
Constructing a triangle using three elements
- Group 1 - construction of a triangle using two sides and the angle between them.
- Group 2 - construction of a triangle using two angles and the side between them.
- Group 3 - construction of a triangle on three sides.
1. segments M 1 N 1 and M 2 N 2.
1. segment MN.
You need to: use a compass and a ruler without scale divisions to construct a triangle.
Segments: M 1 N 1, M 2 N 2, M 3 N 3
You need to: use a compass and a ruler without scale divisions to construct a triangle.
Construct a triangle using two sides and the angle between them
Igor Zhaborovsky © 2011
UROKI MATHEMATICS .RU
Construction
Construction algorithm
1. Let's draw a straight line A .
2. Put it on it using
compass segment AB, equal
segment M 1 N1 .
3. Construct an angle TO YOU, equal
this angle hk .
4. On the beam AM put aside the segment
AC, equal to the segment M 2 N 2 .
5. Let's draw a segment B.C. .
6. Constructed triangle
ABC- sought after.
Construct a triangle using a side and two adjacent angles
Igor Zhaborovsky © 2011
UROKI MATHEMATICS .RU
Construction algorithm
1 . Let's draw a beam AK with the beginning
at the point A .
2 From the beginning of the ray we will postpone
line segment AB, equal to the segment M 1N1 .
3. Let us postpone from the beginning of the ray from
using a compass angle C1AB ,
equal to angle hk .
4. Construct an angle ABC2, equal
corner mn .
5. Point of intersection of rays
AC1 And BC2 denote by a dot WITH .
6. Constructed triangle
ABC- sought after.
Construction
We quickly got up from our desks
And they walked on the spot
- And now we smiled
- Higher, higher we reached.
Straighten your shoulders
raise, lower,
Turn to the left, turn to the left.
And sit down at your desk again.
Construct a triangle using its three sides
Igor Zhaborovsky © 2011
UROKI MATHEMATICS .RU
Construct a triangle using its three sides
Construction algorithm
1. Let's draw a straight line A .
2. Using a compass, draw a segment on it AB, equal to the segment M 1N1 .
3. Construct a circle with
center A and radius M 2 N 2 .
4. Construct a circle with
center IN radius M 3 N 3 .
5. Let us denote one of the intersection points of these circles
dot WITH .
6. Let's draw segments AC And Sun .
7. Constructed triangle ABC- sought after.
Igor Zhaborovsky © 2011
UROKI MATHEMATICS .RU
Task (on one's own)
Construct a triangle using its three sides
Construction algorithm
1. Let's draw a straight line A .
2. Using a compass, draw a segment on it OD= 4 cm
3. Construct a circle with
center ABOUT and radius OE = 2 cm.
4. Construct a circle with
center D and radius DE = 3 cm.
5. Let us denote one of the intersection points of these circles
dot E .
6. Let's draw segments OE And DE .
7. Constructed triangle
OED- sought after.
Given: OD = 4 cm,
DE = 3 cm,
EO = 2 cm.
Igor Zhaborovsky © 2011
UROKI MATHEMATICS .RU
- P. 38 p.84 (learn the memo)
- No. 291 (a,b)
- Problem 1: on a given ray, from its beginning, lay off a segment equal to the given one.
- Solution.
- Let us depict the figures given in the problem statement: ray OS and segment AB.
- Then, using a compass, we construct a circle of radius AB with center O. This circle will intersect the ray OS at some point D.
- The segment OD is the required one.
- Task 2: subtract an angle from a given ray equal to a given one.
- Solution.
- Let us draw the figures given in the condition: an angle with vertex A and a ray OM.
- Let us draw a circle of arbitrary radius with its center at vertex A of the given angle. This circle intersects the sides of the angle at points B and C.
- Then we draw a circle of the same radius with the center at the beginning of this ray OM. It intersects the ray at point D. After this, we construct a circle with center D, the radius of which is equal to BC. Circles intersect at
- two points. Let's denote one
- letter E. We get the angle MOE
- Construct a triangle using two sides and the angle between them. Solution:
- First of all, let us clarify how this problem should be understood, i.e., what is given here and what needs to be constructed.
- Given segments P1Q1, P2Q2 angle hk.
- P1 Q1
- P2 Q2 h
- It is required, using a compass and a ruler (without scale divisions), to construct a triangle ABC whose two sides, say AB and AC, are equal to the given segments P1Q1
- and Р2Q2, and the angle A between these sides is equal to the given angle hк.
- Let's draw a straight line a and on it, using a compass, plot a segment AB equal to the segment P1Q1
- Then we will construct the angle BAM equal to the given angle hк. (we know how to do this).
- On the ray AM we plot a segment AC equal to the segment P2Q2 and draw a segment BC.
- In fact, according to the construction, AB = P1Q1, AC = P2Q2, A = hк.
- The constructed triangle ABC is the desired one.
- In fact, by construction AB = P1Q1, AC = P2Q2,
- A=hк.
- The described construction process shows that for any given segments P1Q1, P2Q2 and a given undeveloped angle hk, the desired triangle can be constructed. Since straight line a and point A on it can be chosen arbitrarily, there are infinitely many triangles that satisfy the conditions of the problem. All these triangles are equal to each other (according to the first sign of equality of triangles), therefore it is customary to say that this problem has a unique solution.
- Construct a triangle using a side and two
- angles adjacent to it.
- P1 Q1
- How was the construction done?
- Does a problem always have a solution?
- Construct a triangle using its three sides.
- Solution.
- Let the segments P1Q1, P2Q2 and P3Q3 be given. It is required to construct a triangle ABC in which
- Let's draw a straight line and, using a compass, plot a segment AB equal to the segment P1Q1. Then we will construct two circles: one with center A and radius P2Q2.,
- and the other with center B and radius P3Q3.
- Let point C be one of the intersection points of these circles. Drawing segments AC and BC, we obtain the required triangle ABC.
- P1 Q1
- P2 Q2
- P3 Q3
- A B A
- Constructing a triangle using three sides.
- The constructed triangle ABC, in which
- AB = P1Q1, AC = P2Q2, BC = P3Q3.
- In fact, by construction AB = P1Q1,
- AC= Р2Q2, BC= Р3Q3, i.e. The sides of triangle ABC are equal to the given segments.
- Problem 3 does not always have a solution.
- Indeed, in any triangle, the sum of any two sides is greater than the third side, therefore, if any of the given segments is greater than or equal to the sum of the other two, then it is impossible to construct a triangle whose sides would be equal to these segments.
- Let's consider the scheme by which construction problems are usually solved using a compass and a ruler.
- It consists of parts:
- 1. Finding a way to solve a problem by establishing connections between the required elements and the data of the problem. Analysis makes it possible to draw up a plan for solving the construction problem.
- 2. Execution of construction according to the planned plan.
- 3. Proof that the constructed figure satisfies the conditions of the problem.
- 4. Study of the problem, i.e. clarifying the question of whether, given any given data, the problem has a solution, and if so, how many solutions.
- Construct a triangle using a side, an adjacent angle, and the bisector of the triangle drawn from the vertex of this angle.
- Solution.
- Required to construct a triangle ABC, which has one of the sides, for example AC, equal to this segment P1Q1, corner A equal to this
- corner hk, and the bisector AD of this triangle is equal to the given
- segment P2Q2.
- Given are the segments P1 Q1 and P2Q2 and the angle hк (Figure a).
- P1 Q1 P2 Q2
- figure a
- Construction (Figure b).
- 1) Let's construct an angle XAU equal to the given angle hk.
- 2) On the ray AC we plot a segment AC equal to this segment P1Q1.
- 3) Construct the bisector AF of the angle XAU.
- 4) On the ray AF we plot a segment AD equal to the given segment P2Q2
- 5) The required vertex B is the point of intersection of the ray AX with the straight line CD. The constructed triangle ABC satisfies all the conditions of the problem: AC = P1Q1,
- A = hк, AD = P2Q2, where AD is the bisector of triangle ABC.
- figure b
- Conclusion: the constructed triangle ABC satisfies all the conditions of the problem:
- AC= P1 Q1 ; A=hk, AD= P2Q2 ,
- where AD is the bisector of triangle ABC