What are the guidelines for proving that three points are collinear? What is the basis for assessing the training results of continuing education students at the lower secondary level in Vietnam?
What are the guidelines for proving that three points are collinear?
Proving that three points are collinear is one of the fundamental and important problems in geometry. To prove this, WWE must demonstrate that the three points lie on the same straight line. There are various methods to achieve this, depending on the assumptions or information provided in the problem.
Students can refer to the following some common methods to prove that three points are collinear:
| A. What is the concept of three collinear points? Three collinear points are three points that lie on the same straight line. B. Relationship of three collinear points - If three points are collinear, then they are distinct and lie on the same straight line. - There is only one and only one straight line passing through any three points. C. Methods to prove three points are collinear (1) Using two adjacent angles where the three points to be proven are on two opposing rays. (2) The three points to be proven lie on the same ray or any line. (3) Two line segments passing through two of the three points to be proven are parallel to a third line. (4) Two lines both passing through two of the three points to be proven are perpendicular to some third line. (5) The straight line passing through two points also passes through the third point. (6) Applying the properties of the bisector of an angle, the perpendicular bisector of a segment, or the properties of the altitudes in a triangle. (7) Applying the properties of a parallelogram. (8) Applying the properties of an inscribed angle. (9) Applying the properties of vertical angles being equal. (10) Proving by contradiction. (11) Proving that the area of the triangle formed by the three points is zero. (12) Applying the properties of the concurrency of line segments. D. Most commonly applied methods to prove three points are collinear Method 1: Applying the property of a straight angle Select any point D: if ∠ABD + ∠DBC = 180 degrees, then the three points A, B, and C are collinear. Method 2: Using Euclid's postulate Given three points A, B, C and a line a. If AB // a and AC // a, then we can assert that the three points A, B, C are collinear. (Based on Euclid's postulate in the Grade 7 Mathematics curriculum) Method 3: Using the property of perpendicular lines If line segment AB ⊥ a and line segment AC ⊥ a, then the three points A, B, and C are collinear. (Theoretical basis of this method: There is only one and only one line a’ passing through point O and perpendicular to the given line a) Or use the property of A, B, C all lying on the perpendicular bisector of a segment (as included in the Grade 7 Mathematics curriculum) Method 4: Using the uniqueness of the angle bisector If two rays OA and OB are the bisectors of angle xOy, then we can assert that the three points O, A, and B are collinear. The theoretical basis for the above method: An angle has only one unique angle bisector. * Alternatively: If two rays OA and OB lie on the same half-plane with boundary containing ray Ox, and if ∠xOA = ∠xOB, then the three points O, A, and B are collinear. Method 5: Using the properties of the perpendicular bisector If K is the midpoint of segment BD, and point K’ is the intersection of segments BD and AC. If point K’ is the midpoint of BD and coincides with K. Then we can conclude that the three points A, K, and C are collinear. (Theoretical basis of this method: Each line segment has only one unique midpoint) Method 6: Using the properties of concurrent lines Prove that three points belong to the concurrent lines of a triangle. For example: Prove point E is the centroid of triangle ABC and line segment AM is the median of angle A, hence the three points A, M, and H are collinear. Additionally, students can apply this to all other concurrent lines of a triangle such as the three altitudes, three angle bisectors, or three perpendicular bisectors in a triangle. Method 7: Using vector method Use the property of two vectors being parallel to demonstrate that a line passes through all three points (i.e., the three points are collinear) Example: Prove vectors AB and AC are parallel, or vectors CA and CB, or vectors AB and BC are parallel, then we can conclude that the three points A, B, and C are collinear. |
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What are the guidelines for proving that three points are collinear? What is the basis for assessing the training results of continuing education students at the lower secondary level in Vietnam? (Image from the Internet)
What is the basis for assessing the training results of continuing education students at the lower secondary level in Vietnam?
Under Clause 1, Article 8 of Circular 43/2021/TT-BGDDT, the basis for assessing the training results of continuing education students at the lower secondary level in Vietnam is as follows:
- Assess the training results of the students based on the requirements to be achieved in terms of main qualities and general competence according to the appropriate levels to the subjects and levels specified in the Continuing Education Program and requirements for specific competencies specified in each subject in the Continuing Education Program.
- Subject teachers, based on the provisions of Point a of Clause 1, Article 8 of Circular 43/2021/TT-BGDDT, comment and assess the training results, progress, outstanding advantages and major limitations of students in the process of regulating training results and learning the subject.
- The class head-teacher shall, based on the provisions of Point a of this Clause, monitor the training results and learning process of the students; refer to comments and assessments of subject teachers, feedback from students' parents, relevant agencies, organizations and individuals in the process of educating students; guide students to self-review; on that basis, comment and assess the training results of the students according to the levels specified in Clause 2 of Article 8 of Circular 43/2021/TT-BGDDT.
What are the regulations on assessing the training results in the whole school year of continuing education students at the lower secondary level in Vietnam?
According to point b, clause 2, Article 8 of Circular 43/2021/TT-BGDDT, the training results in the whole school year of continuing education students at the lower secondary level in Vietnam are assessed into four levels as follows:
- Good: Semester II is assessed as Good, Semester I is assessed at Fair or higher.
- Fair: Semester II is assessed at Fair level, Semester I is assessed at Passed level or higher; Semester II is assessed at Passed, Semester I is assessed at Good; Semester II is assessed at Good, Semester I is assessed at Passed or Failed.
- Passed: Semester II is assessed at Passed, Semester I is assessed at Good, Passed or Failed; Semester II is assessed at Fair, Semester I is assessed at Failed.
- Failed: The remaining cases.
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