16
Figures with an axis of symmetry
Textbook: pp. 56–60
GoalRecognize figures with an axis of symmetry and count their axes.
New words
symmetric figure · simmetrik shaklperpendicular bisector · o‘rta perpendikulyarangle bisector · burchak bissektrisasiregular polygon · muntazam ko‘pburchak
Explanation
If a figure maps onto itself under reflection in a line l, then l is called an axis of symmetry and the figure has axial symmetry. A figure may have one, two, three or even infinitely many axes. A segment is symmetric about its perpendicular bisector. Theorem: the line containing the bisector of an angle is an axis of symmetry of that angle. From this, the line of the apex-angle bisector of an isosceles triangle is its axis of symmetry; an equilateral triangle has three axes and a regular hexagon has six. Every line through the centre of a circle is an axis of symmetry of the circle.
Worked examples
A square has 4 axes of symmetry: two through the midpoints of opposite sides and two along the diagonals.
In isosceles triangle ABC with AB = BC, the bisector BD of angle B is an axis of symmetry. A and C swap, so ∠A = ∠C.
Class activity
“Letter symmetry”: among printed capital letters (A, B, H, M, T, Z, …) find those with an axis of symmetry and draw the axes.
Practice
1
How many axes of symmetry does a regular hexagon have?
6
2
How many axes of symmetry does a regular pentagon have?
5
3
How many axes of symmetry does a rectangle that is not a square have?
2
4
In an isosceles triangle, why does the line of the apex bisector carry vertex A onto vertex C?
That line swaps the two sides of the angle, and since BA = BC, point A lands on C.