AXIAL SYMMETRY: PROPERTIES, EXAMPLES AND EXERCISES - MATHS - 2024

The axial symmetry is when the points of a figure coincide with the points of another figure by a straight bisector called axis of symmetry. It is also called radial, rotational, or cylindrical symmetry.

It is usually applied in geometric figures, but it is easily observable in nature, since there are animals such as butterflies, scorpions, ladybugs or humans that present axial symmetry.

Axial symmetry is exhibited in this photo of the Toronto city skyline and its reflection in the water. (Source: pixabay)

How to find axial symmetric

To find the axial symmetry P 'of a point P with respect to a line (L), the following geometric operations are performed:

1.- The perpendicular to the line (L) that passes through point P.

2.- The interception of the two lines determines a point O.

3.- The length of the segment PO is measured, then this length is copied onto the line (PO) starting from O in the direction from P to O, determining the point P '.

4.- Point P 'is the axial symmetric of point P with respect to the axis (L), since the line (L) is the bisector of the segment PP', being O the midpoint of said segment.

Figure 1. Two points P and P 'are axially symmetric to an axis (L) if said axis is a bisector of the segment PP'

Properties of axial symmetry

- Axial symmetry is isometric, that is, the distances of a geometric figure and its corresponding symmetry are preserved.

- The measure of an angle and its symmetric are equal.

- The axial symmetry of a point on the axis of symmetry is the point itself.

- The symmetric line of a line parallel to the axis of symmetry is also a line parallel to said axis.

- A secant line to the axis of symmetry has as a symmetric line another secant line that, in turn, intersects the axis of symmetry at the same point on the original line.

- The symmetric image of a line is another line that forms an angle with the axis of symmetry of the same measure as that of the original line.

- The symmetric image of a line perpendicular to the axis of symmetry is another line that overlaps the first one.

- A line and its axial symmetric line form an angle whose bisector is the axis of symmetry.

Figure 2. Axial symmetry preserves distances and angles.

Examples of axial symmetry

Nature exhibits abundant examples of axial symmetry. For example, you can see the symmetry of faces, insects such as butterflies, the reflection on calm water surfaces and mirrors or the leaves of plants, among many others.

Figure 3. This butterfly exhibits near perfect axial symmetry. (Source: pixabay)

Figure 4. This girl's face has axial symmetry. (Source: pixabay)

Axial symmetry exercises

Exercise 1

We have the triangle of vertices A, B and C whose Cartesian coordinates are respectively A = (2, 5), B = (1, 1) and C = (3,3). Find the Cartesian coordinates of the triangle symmetric about the Y axis (ordinate axis).

Solution: If a point P has coordinates (x, y) then its symmetric about the ordinate axis (Y axis) is P '= (- x, y). In other words, the value of its abscissa changes sign, while the value of the ordinate remains the same.

In this case, the symmetric triangle with vertices A ', B' and C 'will have coordinates:

A '= (- 2, 5); B '= (- 1, 1) and C' = (- 3, 3) as can be seen in figure 6.

Figure 6. If a point has coordinates (x, y), its symmetric with respect to the Y axis (ordinate axis) will have coordinates (-x, y).

Exercise 2

With reference to triangle ABC and its symmetric A'B'C 'from exercise 1, check that the corresponding sides of the original triangle and its symmetric one have the same length.

Solution: To find the distance or length of the sides we use the Euclidean distance formula:

d (A, B) = √ ((Bx - Ax) ^ 2 + (By - Ay) ^ 2) = √ ((1-2) ^ 2 + (1-5) ^ 2) = √ ((- 1) ^ 2 + (-4) ^ 2) = √ (17) = 4.123

The length of the corresponding symmetrical side A'B 'is calculated below:

d (A ', B') = √ ((Bx'-Ax ') ^ 2 + (By'-Ay') ^ 2) = √ ((- 1 + 2) ^ 2 + (1-5) ^ 2) = √ ((1) ^ 2 + (-4) ^ 2) = √ (17) = 4.123

In this way, it is verified that axial symmetry preserves the distance between two points. The procedure can be repeated for the other two sides of the triangle and its symmetric to check the invariance in length. For example -AC- = -A'C'- = √5 = 2,236.

Exercise 3

In relation to triangle ABC and its symmetric A'B'C 'from Exercise 1, check that the corresponding angles of the original triangle and its symmetric have the same angular measure.

Solution: To determine the measures of the angles BAC and B'A'C ', we will first calculate the scalar product of the vectors AB with AC and then the scalar product of A'B' with A'C '.

Remembering that:

A = (2, 5), B = (1, 1) and C = (3,3)

A '= (- 2, 5); B '= (- 1, 1) and C' = (- 3, 3).

It has:

AB = <1-2, 1-5> and AC = <3-2, 3-5>

similarly

A'B ' = <-1 + 2, 1-5> and AC = <-3 + 2, 3-5>

Then the following scalar products are found:

AB⋅AC = <-1, -4> ⋅ <1, -2> = -1⋅1 + (-4) ⋅ (-2) = -1 + 8 = 7

Similarly

A'B'⋅A'C ' = <1, -4> ⋅ <-1, -2> = 1⋅ (-1) + (-4) ⋅ (-2) = -1 + 8 = 7

The measure of the angle BAC is:

∡BAC = ArcCos (AB⋅AC / (- AB- ⋅- AC-)) =

ArcCos (7 / (4,123⋅2,236)) = 40.6º

Similarly, the measure of angle B'A'C 'is:

∡B'A'C '= ArcCos (A'B'⋅A'C' / (- A'B'- ⋅- A'C'-)) =

ArcCos (7 / (4,123⋅2,236)) = 40.6º

Concluding that axial symmetry preserves the measure of the angles.

Exercise 4

Let a point P be of coordinates (a, b). Find the coordinates of its axial symmetry P 'with respect to the line y = x.

Solution: We will call (a ', b') the coordinates of the symmetric point P 'with respect to the line y = x. The midpoint M of the segment PP 'has coordinates ((a + a') / 2, (b + b ') / 2) and is also on the line y = x, so the following equality holds:

a + a '= b + b'

On the other hand, the segment PP 'has slope -1 because it is perpendicular to the line y = x with slope 1, so the following equality holds:

b - b '= a' -a

Solving for the two previous equalities a 'and b' it is concluded that:

a '= by that b' = a.

That is, given a point P (a, b), its axial symmetry with respect to the line y = x is P '(b, a).

References

1. Arce M., Blázquez S and others. Transformations of the plane. Recovered from: educutmxli.files.wordpress.com
2. Calculation cc. Axial symmetry. Recovered from: calculo.cc
3. Superprof. Axial symmetry. Recovered from: superprof.es
4. wikipedia. Axial symmetry. Recovered from: es.wikipedia.com
5. wikipedia. Circular Symmetry. Recovered from: en.wikipedia.com