Twój koszyk jest pusty
Axially symmetric polygons and the circle
The kite
Reflect triangle \latex{ ABC } across the axis containing side \latex{ AC }.
\latex{ A } and \latex{ C } stay unchanged because they are on the axis of reflection. Mark the reflection of \latex{ B } as \latex{ D }.
Connecting point D to points A and C forms a quadrilateral ABCD, in which the two pairs of adjacent sides are of equal length.
\latex{A}
\latex{B}
\latex{D}
\latex{C}
\latex{l}
A quadrilateral with two pairs of adjacent sides of equal length is called a kite.
\latex{a}
\latex{a}
\latex{a}
\latex{a}
\latex{b}
\latex{b}
\latex{b}
\latex{b}
\latex{c}
\latex{c}
\latex{c}
\latex{c}
\latex{a}
\latex{a}
\latex{a}
\latex{a}
Properties of the kite
- A kite is an axially symmetric quadrilateral.
- The axis of symmetry bisects the angles at its vertices.
- It has two equal angles.
- One diagonal lies along the axis of symmetry.
- Its axis of symmetry bisects the other diagonal perpendicularly.
\latex{\large\alpha}
\latex{\large\alpha}
\latex{\large\beta}
\latex{\large\beta}
\latex{\large\beta}
\latex{\large\beta}
\latex{\large\gamma}
\latex{\large\gamma}
\latex{\large a}
\latex{\large a}
\latex{\large a}
\latex{\large a}
\latex{\large b}
\latex{\large b}
\latex{\large b}
\latex{\large b}
True statements about kites
- There are convex kites.
- There are concave kites.
- Some kites have two axes of symmetry.
- Some kites have diagonals that bisect each other.
- A kite with four axes of symmetry is a square.
The rhombus
Let’s examine what type of kite you get when an isosceles triangle \latex{ ABC } is reflected across the axis containing its base \latex{ AC }.
\latex{ A } and \latex{ C } stay unchanged because they are on the axis of reflection. Mark the reflection of \latex{ B } as \latex{ D }.
All sides of quadrilateral \latex{ ABCD } are of equal length.
\latex{A}
\latex{C}
\latex{l}
\latex{D}
\latex{B}
A quadrilateral with all sides of equal length is called a rhombus.
Properties of the rhombus
- Properties of the rhombus
- Opposite sides are parallel.
- Axes of symmetry bisect the angles.
- Opposite angles are equal.
- Both diagonals lie on the axes of symmetry.
- Diagonals are perpendicular to each other.
- Diagonals bisect each other.
\latex{\alpha}
\latex{\alpha}
\latex{\beta}
\latex{\beta}
\latex{a}
\latex{a}
\latex{a}
\latex{a}
Kites
Rhombuses
Squares
True statements about the rhombus
- Every rhombus is convex.
- Every rhombus is a kite.
- Every square is a rhombus.
- A rhombus with diagonals of equal length is called a square.
The trapezoid
Cut quadrilaterals out of a piece of paper as in the image below.
A quadrilateral with at least one pair of parallel sides is called a trapezoid.
Terminology
The parallel sides of a trapezoid are called the bases. The other two sides are called the legs.
\latex{D}
\latex{C}
\latex{B}
\latex{A}
base
base
leg
leg
The isosceles trapezoid
Imagine a line \latex{ l } and a segment \latex{ AB } that does not intersect \latex{ l }.
Reflect \latex{ AB } across \latex{ l }. Mark the reflection of \latex{ A } as \latex{ D } and the reflection of \latex{ B } as \latex{ C }. Connect the four points.
Since segments \latex{ AD } and \latex{ BC } are perpendicular to \latex{ l }, they are parallel.
\latex{A}
\latex{D}
\latex{C}
\latex{B}
\latex{\alpha}
\latex{\beta}
\latex{\alpha}
\latex{\beta}
\latex{l}
\latex{\begin{rcases}AD \perp l \\BC \perp l\end{rcases}\Rightarrow AD \parallel BC}
The \latex{ ABCD } quadrilateral is a trapezoid symmetrically mirrored along \latex{ axis\,l }. The angles on the same base are equal and mirror images of each other.
A trapezoid with two pairs of equal adjacent angles is called an isosceles trapezoid.
\latex{c}
\latex{a}
\latex{b}
\latex{b}
\latex{a}
\latex{a}
\latex{d}
\latex{d}
\latex{a}
\latex{a}
\latex{a}
\latex{a}
Properties of the isosceles trapezoid
- An isosceles trapezoid is axially symmetric, with the axis of symmetry \latex{ l } bisecting the bases.
- The two legs of the isosceles trapezoid are equal in length.
- The diagonals of the isosceles trapezoid are of equal length and intersect each other on the axis of symmetry.
\latex{\beta}
\latex{l}
base
base
leg
leg
\latex{\beta}
\latex{\alpha}
\latex{\alpha}
Isosceles trapezoids
Rectangle
Squares
True statements about the isosceles trapezoid
- All isosceles trapezoids are convex.
- They can have right angles.
- The bases can be of equal length.
- The diagonals may bisect each other.
- The axis of symmetry can pass through the vertices.
Axially symmetric quadrilaterals
Axially symmetric quadrilaterals can be grouped based on their axes of symmetry.
\latex{1.}
\latex{2.}
\latex{3.}
\latex{4.}
\latex{5.}
\latex{6.}
has an axis of symmetry passing
through a vertex
through a vertex
has an axis of symmetry not
passing through a vertex
passing through a vertex
kite
rhombus
square
rectangle
isosceles
trapezoid
trapezoid
kites
isosceles trapezoids
Kites
Isosceles
trapezoids
trapezoids
Squares
Some quadrilaterals are both kites and isosceles trapezoids. A square is such a quadrilateral.
Regular polygons
Construct a set diagram of rhombuses and rectangles.
What are the properties of the quadrilateral that is part of both sets?
Rhombuses
Rectangle
Squares
Although the sides are of equal length, it is not a square.
In a rhombus, all sides are of equal length, and in a rectangle, all angles are equal.
A quadrilateral with \latex{ 4 } sides of equal length and equal angles is called a square.
A triangle with all sides of the same length and all angles equal is called an equilateral triangle.
Polygons with sides of equal length and equal angles are called regular polygons.
regular rectangle
(square)
(square)
regular
triangle
triangle
regular
pentagon
pentagon
regular
hexagon
hexagon
regular
heptagon
heptagon
regular
octagon
octagon
Every regular polygon is axially symmetric. A regular triangle has \latex{ 3 } axes of symmetry, a square has \latex{ 4 }, a regular pentagon has \latex{ 5 }, and a hexagon has \latex{ 6 }.
The axes of symmetry of regular polygons intersect at a single point. From this point, every vertex of the regular polygon is equidistant. Therefore, if we draw a circle centred at the intersection of the axes of symmetry, with a radius equal to the distance from this point to one of the vertices, we can see that all vertices lie on the circle. This circle is called the circumscribed circle of the polygon.
\latex{r}
\latex{r}
\latex{r}
\latex{r}
\latex{60°}
The circle
Draw any circle and any line in a plane. How many common points can the circle and the line have?
There are three possibilities:
\latex{1.}
\latex{k}
\latex{e}
\latex{2.}
\latex{k}
\latex{e}
\latex{E}
tangent
\latex{3.}
\latex{k}
\latex{B}
\latex{A}
\latex{e}
secant
- Circle \latex{ k } and line \latex{ e } have no points in common.
- Circle \latex{ k } and line \latex{ e } intersect at one point. Line e is tangent to circle \latex{ k }, and their common point is the point of tangency \latex{ E }.
- Circle \latex{ k } and line \latex{ e } intersect at two points. In this case, the line is a secant of the circle.
\latex{O}
\latex{\large r}
\latex{\large d}
\latex{\large k}
\latex{\large i}
\latex{O}: center;
\latex{r}: radius;
\latex{k}: circle;
\latex{d}: diameter;
\latex{r}: radius;
\latex{k}: circle;
\latex{d}: diameter;
\latex{i}: arc.
The part of the secant that lies inside the circle is called the chord. A chord is a line segment that connects two arbitrary points on the circle.
True statements about the chord
- Every circle has an infinite number of chords.
- The longest chord of a circle is its diameter.
- The perpendicular bisector of a chord passes through the centre of the circle.
- The perpendicular bisector of a chord also bisects the angle formed by the radius lines drawn to the endpoints of the chord.
Reasoning:
Since \latex{OA=OB=r}, triangle \latex{ OAB } is an isosceles triangle, and its axis of symmetry is the line \latex{ l }.
The term isosceles trapezoid comes from the fact that you can draw a circumscribed circle around it. All the vertices of the trapezoid will lie on the circle, and the sides of the trapezoid will become the chords of the circle.
\latex{O}
\latex{O}
\latex{O}
\latex{A}
\latex{B}
\latex{\alpha}
\latex{r}
\latex{h}
\latex{e}
\latex{h}: chord, \latex{AB}: chord
\latex{h_1}
\latex{h_2}
\latex{h_3}
\latex{d = h_4}
\latex{r}
\latex{r}
\latex{B}
\latex{A}
\latex{k}
\latex{l}
Exercises
{{exercise_number}}. Fold a sheet of paper in half and cut out the following figures along the fold. Which shape will make an isosceles trapezoid?
fold
\latex{1.}
\latex{2.}
\latex{3.}
\latex{4.}
\latex{5.}
{{exercise_number}}. Can an isosceles triangle be divided into two parts by a straight line so that one part is an isosceles triangle and the other is an isosceles trapezoid?
{{exercise_number}}. The bases of an isosceles trapezoid are \latex{ 6\,cm } and \latex{ 8\,cm }, and both legs are \latex{ 7\,cm } long. What is its perimeter?
{{exercise_number}}. The perimeter of an isosceles trapezoid is \latex{ 30\,cm }. One of its bases is \latex{ 6\,cm }, and both legs are \latex{ 7\,cm } long. What is the length of the other base?
{{exercise_number}}. Are the following statements true or false?
- The diagonals of every isosceles trapezoid bisect each other.
- Some isosceles trapezoids have exactly two axes of symmetry.
- There are trapezoids with all sides of equal length.
- There is no isosceles trapezoid whose diagonals are perpendicular to each other.
{{exercise_number}}. Draw an isosceles trapezoid whose bases are the same length. What kind of quadrilateral did you get?
{{exercise_number}}. Draw an isosceles trapezoid that fulfils all the following conditions:
- All sides are equal in length.
- All angles are equal.
- The diagonals are equal in length.
- The diagonals are perpendicular, but the trapezoid is not a square.
{{exercise_number}}. Fold a sheet of paper in half and cut out the following figures along the fold. Which shape will make a kite?
fold
\latex{1.}
\latex{2.}
\latex{3.}
\latex{4.}
\latex{5.}
\latex{6.}
{{exercise_number}}. A kite has two sides measuring \latex{ 6 } \latex{ cm } and \latex{ 8 } \latex{ cm }. What is its perimeter?
{{exercise_number}}. A kite has two adjacent sides measuring \latex{ 6.4 } \latex{ cm } and \latex{ 8.7 } \latex{ cm }. What is its perimeter?
{{exercise_number}}. Draw a kite with all sides measuring \latex{ 5 } \latex{ cm }. How many such kites can you draw? How many axes of symmetry do they have?
{{exercise_number}}. Are the following statements true or false?
- Every kite has two equal angles.
- The diagonals of every kite bisect each other.
- The kite's axis of symmetry splits two of its angles in half.
- Some kites are also rhombuses.
- The perimeter of a kite is twice the sum of the lengths of two adjacent sides.
{{exercise_number}}. What is the perimeter of a rhombus whose sides are all \latex{ 6 } \latex{ cm } long?
{{exercise_number}}. How long are the sides of a rhombus with a perimeter of \latex{ 30 } \latex{ cm ?}
{{exercise_number}}. The perimeter of a rhombus is \latex{ 6 } \latex{ cm } longer than the length of one of its sides. How long are its sides? What is the perimeter of the rhombus?
{{exercise_number}}. Draw a rhombus with all sides measuring \latex{ 5 } \latex{ cm }. How many such rhombuses can you draw?
{{exercise_number}}. Are the following statements true or false?
- Some rhombuses have four axes of symmetry.
- Every rhombus has a right angle.
- If the length of one side of a rhombus is a, then its perimeter is \latex{ 4\times a }.
{{exercise_number}}. Someone has drawn the axes of symmetry for various quadrilaterals. However, some are wrong. Identify which lines are not a proper axis of symmetry.
\latex{ A) }
\latex{ B) }
\latex{ C) }
\latex{ D) }
\latex{1.}
\latex{2.}
\latex{4.}
\latex{3.}
\latex{1.}
\latex{2.}
\latex{4.}
\latex{3.}
\latex{1.}
\latex{2.}
\latex{4.}
\latex{3.}
\latex{1.}
\latex{3.}
\latex{2.}
Quiz
Use four congruent isosceles triangles to make as many quadrilaterals as possible.
