Math 6 Chapter 1 Lesson 10: Midpoint of line segment
1. Theory
The midpoint of line segment AB is the point M on that line and is equidistant from two points A and B.
We have:
M is the midpoint of line segment AB \( \Leftrightarrow \left\{ \begin{array}{l}M \in AB\\MA = MB\end{array} \right.\)
Or
M is midpoint of line segment AB \( \Leftrightarrow \left\{ \begin{array}{l}AM + MB = AB\\MA = MB\end{array} \right.\)
Or
M is the midpoint of line segment AB \( \Leftrightarrow AM = MB = \frac{1}{2}AB.\)
2. Illustrated exercise
Question 1: On the ray Ox there are three points A, M, B. Knowing OA = 8, OB = 14 and OM = 11. Show that M is the midpoint of line segment AB.
Solution guide:
We have OA < OM < OB \( \Rightarrow \) Point M on line AB (1)
We have MA=OMOA= 3 again; MB=OBOM= 3 \( \Rightarrow \) MA = MB (2)
From (1) and (2) deduce dcm.
Verse 2: On the xray there are three points A, B, and C knowing OA = 10cm, OB = 24cm, OC = 16cm. Let MN in order be the midpoints of the line segments AC, BC.
a) Prove that point C lies on line segment AB.
b) Calculating OM, ON infers point C on line segment MN.
c) Calculate MN.
Solution guide:
a. We have OA < OC < OB, so C lies between two points A and B.
So C belongs to the line segment AB.
b. We have: AC=OCOA=1610=6 (cm)
Since M is the midpoint of AC, \(MA = MC = \frac{{AC}}{2} = 3\,\,(cm)\)
So OM=OA+AM=10+3=13 (cm).
Similarly, we have: BC=OBOC=2416=8 (cm)
Since N is the midpoint of BC, we have: \(NC = NB = \frac{{BC}}{2} = 4\,\,(cm)\)
So ON=OC+CN=16+4=20 (cm).
Since OM < OC < ON, C lies between two points M and N.
c. We have: MN=MC+CN=4+3= 7 (cm).
3. Practice
3.1. Essay exercises
Question 1: Let M be the midpoint of line segment AB and C be any point between A and M. Prove that: \(CM = \frac{{CB – CA}}{2}.\)
Verse 2: On the line xy give three points A, B, C in that order. Let M and N be the midpoints of AB and BC respectively. Prove that: \(MN = \frac{{AB + BC}}{2}.\)
3.2. Multiple choice exercises Bài
Question 1: M is the midpoint of line segment AB if and only if
A. MA = MB
B. AM = 1/2 AB
C. MA + MB = AB
D. MA + MB = AB and MA = MB
Verse 2: If we have P as mid point of MN then
A. \(MP = NP = \frac{{MN}}{2}\)
B. MP + NP = 2MN
C. \(MP = NP = \frac{{MN}}{4}\)
D. MP = NP = MN
Question 3: Given line segment AB 12cm long, M is midpoint of line segment AB. That warehouse, the length of the line segment MA is equal to
A. 3cm
B. 15cm
C. 6cm
D. 2cm
Question 4: Let I be the midpoint of line segment MN. We know NI = 8cm. Then, the length of the line segment MN is equal to
A. 4cm
B. 16cm
C. 21cm
D. 24cm
Question 5: Let O lie on the line xy. On the xray take a point A such that OA = 5cm. On ray Oy take point B such that OB = 6cm. Let I and K be the midpoints of OA and OB respectively. Calculate IK.
A. 4cm
B. 4.5cm
C. 5cm
D. 5.5cm
Question 6: On the xray, take two points A and B such that OA=3cm, OB=6cm. Choose the wrong sentence
A. Point A is between two points O and B
B. Point A is the midpoint of segment OB
C. Point O is the midpoint of segment AB
D. OA = AB = 3cm
Verse 7: Given line segment AB. Let M and N be the midpoints of the lines AB and AM respectively. Assume AN = 1.5cm. What is the length of line segment AB?
A. 1.5cm
B. 3cm
C. 4.5cm
D. 6cm
Verse 8: Let line segment AB = 8cm. Let I and K be the midpoints of the lines AB and AI respectively. What is the length of the line segment IK?
A. 8cm
B. 4cm
C. 2cm
D. 6cm
Verse 9: Let line segment AB = 14cm. On the ray AB take a point M such that AM = 7cm. Choose the wrong sentence.
A.M lies between A and B
B. AM = BM = 7cm
C. BM = AB
D.M is the midpoint of AB
Question 10: On the Xray there are points A, B such that OA = 2cm, OB = 5cm. Let M be the midpoint of line segment OB. Calculate the length of the line segment AM
A. 1.5cm
B. 0.5cm
C. 1cm
D. 2cm
Question 11: Let line segment AB = 2a. Point O lies between two points A and B. Let M and N be the midpoints of OA and OB, respectively. The length of line segment MN is:
A. 2a
Three
C. \(\frac{{3{\rm{a}}}}{2}\)
D. 0.5a
4. Conclusion
Through the lesson The Midpoint of this line, you need to complete some of the objectives given by the lesson, such as:

Understand the concept and properties of the midpoint of a line segment.

Apply related exercises.
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