Math 8 Chapter 1 Lesson 6: Axis symmetry
1. Theoretical Summary
1.1. Two points are symmetrical about the line
Define: Two points are symmetric about the line d if d is the perpendicular bisector of the line joining the two points.
Eg: A and B are symmetric about d \( \Leftrightarrow \) d is the orthogonal of AB
If the point \(M \in d\) then the point symmetrical to M through d is also the point M.
1.2. Two figures that are symmetrical about each other through a straight line
a. Define: Two figures are said to be symmetric about the line d if each point in one is symmetric about d to a point in the other and vice versa.
b. Nature:
Theorem: If two line segments AB and A’B’ have points A and A’, B and B’ that are symmetric about line d, then those two segments are equal and symmetrical about line d.
c. Attention: We have
Symmetrical figure through a line d of:
– A straight line is a straight line
– A line segment is a line segment
– An angle is an angle equal to it
– A triangle is a triangle equal to it.
– A circle is a circle whose radius is equal to the radius of the given circle.
1.3. Axis of symmetry of a figure
a. Define: The line d is said to be the axis of symmetry of the figure F if the point of symmetry of each point of each of the figures F through the line d is also in the figure F.
b. Some familiar symmetry axes
– A line segment whose axis of symmetry is the perpendicular bisector of that line.
An angle whose axis of symmetry is the bisector of the angle.
– Two intersecting lines whose axis of symmetry is two lines containing the bisectors of the angles formed by the two lines; These two axes of symmetry are perpendicular to each other.
– An isosceles triangle has an axis of symmetry, the altitude is also the bisector, the median, and the base. An equilateral triangle has three axes of symmetry.
An isosceles trapezoid whose axis of symmetry is the line passing through the midpoints of the two bases.
2. Illustrated exercise
Question 1: Given two points A and B lying in opposite halfplanes, the edge is a given line a. Find on the line a a point such that the difference between the distances from M to two points A and B has the greatest value.
Solution guide
Let B’ be the point symmetrical to the point B through the line a.
Join AB’ the line AB’ intersects the line a at the point M. That is the point M to find.
Since MB = MB’ and three points A, B’, M are collinear, so:
MA – MB = MA – MB’ = AB’
We show that for every point \(M’ \in a\) where \(M’ \ne M\) is different:
M’A – M’B > MA – MB
Indeed, we have M’B’ = M’B, so:
M’A – M’B = M’A – M’B’
In triangle AM’B’, according to the triangle inequality, we have:
\(\left {M’A – M’B’} \right \le \left {MA – MB} \right\)
The string “=” occurs only if M’ coincides with M.
Attention: In a special case, when AB’ is parallel to a, that is, two points A and B are equidistant from the line a, then the point M to find is the intersection of AB with the line a.
Verse 2: Given an acute angle xOy and a point M in the outer region of that angle.
Call: \({M_1}\) is the symmetry point of M about Ox
\({M_2}\) is the symmetry point of M about Oy
I is the midpoint of the line segment \({M_1}{M_2}.\)
a. Determine the axis of symmetry through which \({M_1}\) and \({M_2}\) are two points of symmetry to each other.
b. The two rays OM and OI are symmetric about which axis?
Solution guide
a. M and \({M_1}\) are symmetric about Ox
\( \Rightarrow O{M_1} = OM\)
M and \({M_2}\) are symmetric about Oy
\( \Rightarrow O{M_2} = OM\)
So \(O{M_1} = O{M_2}\)
\( \Rightarrow \Delta O{M_1}{M_2}\) balance;
So OI is the perpendicular bisector of \({M_1}{M_2}.\)
b. Draw the bisector Oz of the angle xOy.
\(\widehat {{M_1}OM} = \widehat {2MOx} = 2(\widehat {MOy} + \widehat {xOy})\)
\(\begin{array}{l}\widehat {{M_2}OM} = 2\widehat {MOy}\\ \Rightarrow \widehat {{M_1}O{M_2}} = 2\widehat {xOy}.\end {array}\)
From here, we have: \(\widehat {{M_1}OI} = \frac{1}{2}\widehat {{M_1}O{M_2}} = \widehat {xOy}\) (1)
We also have \(\widehat {{M_1}Ox} = \widehat {MOx}\) (2)
From (1) and (2) deduce \(\widehat {IOx} = \widehat {MOy}\) (3)
Since Oz is the bisector of xOy, \(\widehat {xOz} = \widehat {yOz}\) (4)
From (3) and (4) we have: \(\widehat {MOz} = \widehat {IOz}\)
So Oz is the bisector of angle \(\widehat {MOI}\) or OM and OI are symmetric about Oz.
Question 3: Let ABC be a triangle perpendicular at vertex A. Draw altitude AH. Let D, E, respectively, be the symmetry points of the point H through AB, AC. Prove that:
a. Point A is the midpoint of line segment DE.
b. DE = 2AH.
Solution guide
a. D and H are symmetrical about each other through AB, so AD = AH.
The triangle DAH is isosceles at vertex A where \(AB \bot DH\) so AB is also the bisector of angle DAH, so: \(\widehat {{A_1}} = \widehat {{A_3}}\)
Similarly, we have AE =AH and \(\widehat {{A_2}} = \widehat {{A_4}}\)
From the above results we have:
\(\widehat {{A_3}} + \widehat {{A_1}} + \widehat {{A_2}} + \widehat {{A_4}} = 2(\widehat {{A_1}} + \widehat {{A_2} ) }) = {2.90^0} = {180^0}\)
\( \Rightarrow \) Three points D, A, E are collinear (1)
We also have AD = AE (2)
From (1) and (2) it follows that A is the midpoint of DE.
b. We have immediately AD = AH and AE = AH
\(\Rightarrow DE = AD + AE = 2AH\)
3.1. Essay exercises
Question 1: Let ABC be a triangle. Line high AH. Let D, E, respectively, be the symmetry points of the point H through the sides AB, AC. The line DE intersects AB, AC at M, N respectively. Prove:
a. Triangle DAE is an isosceles triangle.
b. HA is the bisector of angle MHN.
c. Three lines BN, CM and AH are concurrent.
d. BN, CM are the altitudes of triangle ABC.
Verse 2: Given an acute angle xOy and a point A in the interior of that angle. Find on the side Ox a point B, on the side Oy a point C such that triangle ABC has the smallest perimeter.
Question 3: Let ABCD square trapezoid (AB // CD)
Call E, F in the order of the symmetry points of point B and point A through the line DC; G; H are respectively the symmetry points of the point C and the point E through the line AD.
a. Prove that point D is the midpoint of the line segments BH.
b. Prove AH // BF and CH // BG.
3.2. Multiple choice exercises
Question 1: Choose the correct statement
A. Two points A and B are symmetrical about the line d if d passes through the midpoint of AB
B. Two points A and B are symmetrical about the line d if d is perpendicular to AB
C. Two points A and B are symmetrical about the line d if d is perpendicular to AB at the midpoint of AB
D. Two points A and B are symmetrical about the line d if d is parallel to AB
Verse 2: Choose the wrong statement
Symmetrical figure through a line d of:
A. A line segment is a line segment that is equal to it
B. A straight line is a line equal to it
C. An angle is an angle equal to it
D. A triangle is a triangle equal to it
Question 3: What is the axis of symmetry of an isosceles trapezoid?
A. Midsegment of an isosceles trapezoid
B. Two diagonals of an isosceles trapezoid
C. The line passing through the midpoints of the two bases
D. The line is perpendicular to the two bases
Question 4: Let segment AB = 5 cm A_{first}REMOVE_{first} Symmetrical to AB through d, length A1B1=?
A. 3 cm
B. 5 cm
C. 10 cm
D. 15 cm
Question 5: Let triangle ABC and triangle A’B’C’ be symmetrical to each other through d, given AB = 6cm, AC = 8 and the perimeter of triangle ABC is 24cm. Ask B’C’ =?
A. B’C’ = 8 cm
B. B’C’ = 10 cm
C. B’C’ = 12 cm
D. B’C’ = 14 cm
4. Conclusion
Through this lesson, you should know the following:

Understand the concept of axial symmetry, shapes with symmetry axes and identify the axis of symmetry.

Remember the properties of axial symmetry, axial symmetry.

Apply knowledge to solve some related problems.
.
=============
Leave a Reply