Math 6 Chapter 2 Lesson 1: Half plane
1. Summary of theory
1.1. Half plane
a) Flat surface
– A table top, a table top, a spread sheet of paper… give us an image of the plane.
– The plane is not constrained in all directions.
b) Half plane
– The figure consisting of the line a and the part of the plane divided by a is called the half plane of the edge a.
– Two halfplanes that have a common edge are called opposite halfplanes.
– In the figure, the shore halfplane a’ contains the point B and the shore halfplane a’ contains the point A is two opposite halfplanes.
– Two points A and C (or A and B) lying on the opposite side of line a or in two opposite halfplanes border a, segment AC (or AB) intersects a.
– Any two points B and C lie on the same side of line a, then segment BC does not intersect a.
Any line lying on a plane is also a common boundary of two opposite halfplanes.
1.2. Three rays with a common origin – The ray lying between two rays
– For three Ox rays; Oy; Original Oz. Take the point M ∈ Ox; N Oy (M; N does not coincide with O)
– If Oz ray intersects line segment MN at a point between M and N, then Oz ray lies between Ox and Oy rays.
2. Illustrated exercise
Question 1:
a) Give other names of the two halfplanes (I), (II).
b) Connect M to N, connect M to P. Does the line segment MN intersect a? Does the line MP cut a?
Solution guide
a) Another name for the two halfplanes (I), (II) is: Plane (I) is the shore plane \(a\) containing the point \(N,\) Plane (II) is the plane edge a does not contain the point \(N.\)
– The line segment \(MN\) does not intersect \(a\).
– The line segment \(MP\) has cut \(a\).
Verse 2: Given a line xy and three points A, B, C not on xy. Knowing the line xy intersects two lines BA and BC.
a) Explain why point A and point C are on the same half plane as the line xy.
b) Does the line xy intersect line AC, why?
Solution guide
a) A and C are on the same side with respect to B with respect to xy.
b) No, because A and C are in the same halfplane of the shore, xy.
Question 3: Given three distinct points A, B, and C and a line xy that does not pass through any of them. Know that xy cuts the line segment AB.
a) Do A and B belong to the same halfplane of the line xy?
b) Why must two of the three points A, B, and C belong to the same halfplane?
c) Why does the line xy have to cut one of the remaining two segments AC or BC?
Solution guide
a) No.
b) If B does not belong to the same half plane of the xy edge, then A and C both belong to the half plane of the edge xy.
c) If A and C are not in the same halfplane of the xy edge, then C and B are also in the opposite halfplane of the halfplane containing A, the edge is the line xy.
3. Practice
3.1. Essay exercises
Question 1: Given a line xy and three points A, B, C that are not on xy. Given that line segment AB does not intersect line xy and line segment AC intersects xy at a point P.
a) Prove that two points B and C lie in two planes opposite to each other, and that the edge is the line xy.
b) Does line segment BC intersect line xy?
Verse 2: Given four line segments AB, BC, CD, DA and a point O as shown in the figure.
a) Show that a line \(\Delta \) passing through O and intersecting one of the four given segments must cut the second line.
b) Is there any line that intersects the four given line segments? If yes, how many lines?
c) Locate the point O so that through O we can draw two lines, each intersecting all four given line segments.
3.2. Multiple choice exercises
Question 1: For drawings
How many pairs of opposite points have a shore?
A. 1
B. 2
C. 3
D. 4
Verse 2: Given three noncollinear points M, N, and P that lie outside the line d. We know that the line d intersects segment MN but not segment MP. Which of the following is the most correct conclusion?
A. Two points M and P lie on the same side of the line d
B. Two points M, N lie on the opposite side of the line d
C. Points N and P belong to two opposite halfplanes
D. Both A, B, C are correct
Question 3: Let the ray Oz lie between two rays Ox and Oy. Ot ray lies between two rays Oz, Oy. Choose the correct conclusion
A. Ot rays lie between two rays Ox and Oz
B. Oz rays lie between two rays Ox, Ot
C. Ox rays lie between two rays Oz, Ot
D. All A, B, C are wrong
Question 4: Choose the best answer
A. The plane is not restricted in all directions
B. The table face is an image of the plane
C. Any straight line is the common edge of two opposite halfplanes
D. All of the above answers are correct
Question 5: Given three common rays Ox, Oy, Oz with A in Ox, I in Oy, K in Oz. If the point K lies between two points A and I, then
A. Ox rays lie between two rays Oz, Oy
B. Oy ray lies between two rays Oz, Ox
C. Oz rays lie between two rays Ox, Oy
D. All A, B, C are wrong
Question 6: For the following picture
Name points on the same halfplane (I) with edge a
A. Two points D, E
B. Two points E, B
C. Two points A and B
D. Two points A, E
4. Conclusion
Through this lesson, you should know the following:

Understand what a plane is, about the ray lying between two other rays.

Recognize halfplanes.

Know how to draw, recognize the ray lying between two rays.
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