Symmetry in mathematics is a fundamental concept that refers to a kind of balance or correspondence between parts of a figure or an equation. It indicates that one part of a figure or equation is a mirror reflection, rotation, or translation of another part.
Symmetry can be found in various branches of mathematics including geometry, algebra, and even complex number theory. In this article, we are going to see a brief introduction the Symmetry and also the Solved Problems and Practice Problems on Symmetry.
There are various types of Symmetry in mathematics:
- Reflective Symmetry (Bilateral Symmetry): This occurs when a figure can be divided into two mirror-image halves by a line, which is known as the line of symmetry.
- Rotational Symmetry: A figure has rotational symmetry if it can be rotated (less than a full circle) about a central point and still look the same as it did before the rotation.
- Translational Symmetry: This type of symmetry occurs when a figure can be moved (translated) along a path in a particular direction and still appear unchanged.
- Glide Reflection Symmetry: Glide reflection symmetry combines a translation and a reflection. A figure shows glide reflection symmetry if it can be reflected across a line and then slid along the same line.
- Scaling Symmetry (Dilational Symmetry): In scaling symmetry, figures or patterns are identical in shape but different in size, often proportionally scaled versions of each other.
Important Formulas in Symmetry
Axis of symmetry for parabola:
- Standard form (y=ax2+bx+c): x = -b/2a
- Vertex form (y=a(x-h)2 + k): x = h (the vertex’s x-coordinate)
Line symmetry equation:
Rotational for a shape with n sides and rotational symmetry:
- 360o/n
Point symmetry formula for a point (x, y) with respect to the origin:
- (-x, -y)
Coordinates of points on a circle with center (h, k) and radius r:
where is the angle of rotation.
Symmetry Question and Solutions
Q1: Determine the order of rotational symmetry for a regular pentagon.
Solution:
A regular pentagon has 5 equal sides and 5 equal angles. The order of rotational symmetry is 5, as rotating the pentagon by 360o/5 = 72o makes it look unchanged.
Q2: Identify the lines of reflectional symmetry for the letter ‘H’.
Solution:
The letter ‘H’ has two lines of reflectional symmetry:
A horizontal line passing through the middle of the letter, reflecting the upper half onto the lower half.
A vertical line passing through the center, reflecting the left side onto the right side.
Q3: Given a function f(x) = x2, determine if it exhibits any symmetry and, describe it.
Solution:
The function f(x) = x2 is symmetric about the y-axis(vertical symmetry). This means that if you reflect the graph of the function across the y-axis, it will look the same.
Mathematically, f(x) = f(-x) for all values of x.
symmetry about the y-axis.
Q4: Find the point of symmetry for the parabola y = 3x2 + 6x + 2.
Solution:
The point of symmetry for a parabola in the form y = ax2 + bx + c, use the formula x=-b/2a.
Here a = 3 and b = -6.
Values into the formula, x = -(-6/2.3) = 1
Therefore, the point of symmetry is (1,f(1)), where f(1) is the corresponding y-coordinate.
Q5: Determine if the triangle with vertices A(1,2), B(3,4), and C(5,2) has any symmetry.
Solution:
Line of reflectional symmetry passing through the midpoint of side BC and perpendicular to BC. The triangle into two congruent halves. Therefore, the triangle has reflectional symmetry.
Q6: Does the equation y = x3 exhibit any symmetry? If so, describe it.
Solution:
The equation y = x3 exhibits point symmetry about the origin, that if we rotate the graph by 180 degrees about the origin, it looks the same.
In Mathematically y = -y for all values of x, indicating point symmetry.
Q7: Determine the order of rotational symmetry for an equilateral triangle.
Solution:
An equilateral triangle has 3 equal sides and 3 equal angles.
The order of rotational symmetry is 3, as rotating the triangle by 360o/3 = 120o makes it unchanged.
Q8: Determine the function f(x) = sin (x) exhibits any symmetry. If so, describe it.
Solution:
The function f(x) = sin (x) exhibits odd symmetry about origin. If we reflect the graph across the origin, it looks unchanged.
Q9: Find the point of symmetry for the graph of y = -2x2 + 4x – 1.
Solution:
Formula x = -b/2a,
Where a = -2 and b= 4
Getting into the formula: x = -4/2 . -2 =1.
Point of Symmetry is (1,f(1)).
Q10: Identify the order of rotational symmetry for a regular octagon.
Solution:
A regular octagon has 8 equal sides and 8 equal angles. The order of rotational symmetry is 8, as rotating the octagon by 360o/8 = 45o makes is unchanged.
Q11: Equation y = 1/x exhibit any symmetry? Describe it.
Solution:
Equation y = 1/x exhibits rotational symmetry of order 2 origin. That if we rotate the graph by 180 degrees about the origin, looks same.
Mathematically -f(-x) = f(x) for all values of x except x = 0.
Rotational symmetry order is 2.
Symmetry Practice Problems
Problem 1: Identify the lines of symmetry for the following shapes?
- Square
- An Equilateral triangle
- Circle
Problem 2: Determine the order of rotational symmetry for each of the following?
- A regular pentagon
- A circle
Problem 3: The following functions exhibit any symmetry? Describe the type of symmetry?
- f(x) = x2
- g(x) = cos (x)
Problem 4: Find the point of symmetry for each of the parabolas?
- y = 5x2 + 10x + 2
- y = -3x2 – 5x + 1
Problem 5: Determine if the quadrilateral with vertices A(1,1), B(3,5), C(7,5), and D(5,1) has any lines of symmetry.
Problem 6: Identify the order of rotational symmetry for the capital letters?
- E
- F
- k
Problem 7: Find the equations of lines of symmetry for the following geometric shapes?
- A rectangle with vertices at (0,0), (0,4), (3,4), and (3,0)
- An isosceles triangle with vertices at (0,0), (4,0), and (2,3)
Problem 8: Determine the lines of symmetry for the parallelogram with vertices at (-1,1), (3,1), (4,4), and (0,4).
Problem 9: Find the point of symmetry for each of the following curves, the graph of y = sin (x)?
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