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Plane figures, polygons
Polygonal figures
Plane figures
Non-polygonal figures
Classification of plane figures
Plane figures have no thickness and are composed of line segments, curves, or the combination of these.
Non-polygonal figures have at least one side composed of curved lines.
Polygonal figures are closed figures composed of only straight lines. Polygons can be classified according to the number of sides or whether they are convex or concave.
Convex plane figures
A plane figure is convex if every point on any line segment connecting two points of the figure lies inside the figure.
Concave figures
A plane figure is concave if there is a line segment connecting two points of the figure that does not entirely lie inside the figure.
Polygons
Line segments bordering polygons are called sides \latex{ (a, b, c, d, e) }.
The point where two sides meet is called a vertex \latex{ (A, B, C, D, E) }; vertices connected by the same side are called adjacent vertices (e.g. \latex{ A } and \latex{ B }). A line segment connecting two non-adjacent vertices is called a diagonal (e.g. diagonal \latex{ BD }).
\latex{ A }
\latex{ E }
\latex{ e }
\latex{ d }
\latex{ D }
\latex{ c }
\latex{ C }
\latex{ b }
\latex{ B }
\latex{ a }
vertex
side
diagonal
diagonal
side
vertex
\latex{ C }
\latex{ B }
\latex{ D }
\latex{ A }
Polygons can be classified based on the number of sides.
triangles
quadrilaterals
pentagons
hexagons
In polygons, the number of vertices is equal to the number of sides.
Example
How many diagonals do the following polygons have?
a) triangle | b) quadrilateral | c) pentagon | d) hexagon |
Solution
- Triangles do not have diagonals because they only have adjacent vertices.
- Quadrilaterals have two diagonals connecting the opposite vertices.
- Drawing and counting all the diagonals of the pentagon, you can see that it has \latex{ 5 } in total.
- Drawing and counting all the diagonals of the hexagon, you can see that it has \latex{ 9 } in total.
Number
of
vertices
of
vertices
Number
of
sides
of
sides
Number
of
diagonals
of
diagonals
triangle
quadrilateral
pentagon
hexagon
\latex{ 3 }
\latex{ 3 }
\latex{ 0 }
\latex{ 4 }
\latex{ 4 }
\latex{ 2 }
\latex{ 5 }
\latex{ 5 }
\latex{ 5 }
\latex{ 6 }
\latex{ 6 }
\latex{ 9 }
Draw the diagonals of a hexagon according to the following method:
Draw all possible diagonals from vertex \latex{ A } \latex{ (3) }. If you draw \latex{ 3 } diagonals starting from vertices \latex{ B, C, D, E } and \latex{ F }, then you can see that there are \latex{ 6 \times 3 = 18 } diagonals connecting \latex{ 6 } vertices. However, every vertex was counted twice, so there are \latex{ 18 \div 2 = 9 } diagonals.
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
This method can also be used in the case of polygons with many sides.
Exercises
{{exercise_number}}. Which of the following plane figures are
a) convex;
b) concave;
c) polygons;
d) concave polygons?
\latex{ 1. }
\latex{ 2. }
\latex{ 3. }
\latex{ 4. }
\latex{ 5. }
\latex{ 6. }
{{exercise_number}}.
a) Draw a convex and a concave polygon.
b) Draw a convex and a concave non-polygonal figure.
b) Draw a convex and a concave non-polygonal figure.
{{exercise_number}}. The drawing shows a dwarf gold mine.
At least how many torches do they need to light all the passages? (One torch can illuminate a passage in all directions.) (→)
{{exercise_number}}. Find on the drawing the
a) vertices of a convex quadrilateral;
b) vertices of a concave quadrilateral;
c) vertices of a convex pentagon;
d) vertices of a concave pentagon. (→)
b) vertices of a concave quadrilateral;
c) vertices of a convex pentagon;
d) vertices of a concave pentagon. (→)
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ G }
\latex{ H }
{{exercise_number}}. Find congruent polygons in the image.
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 11 }
\latex{ 12 }
\latex{ 13 }
\latex{ 14 }
\latex{ 15 }
\latex{ 16 }
{{exercise_number}}. Use the letters of the vertices to designate the sides and diagonals of the following polygons.
a)
\latex{ B }
\latex{ A }
\latex{ C }
\latex{ D }
b)
\latex{ B }
\latex{ A }
\latex{ C }
\latex{ D }
c)
\latex{ B }
\latex{ A }
\latex{ C }
\latex{ D }
\latex{ E }
{{exercise_number}}. Draw a convex
a) triangle;
b) quadrilateral;
c) pentagon;
d) hexagon;
e) heptagon.
Draw the diagonals starting from one of the vertices with a blue pencil. How many are there?
{{exercise_number}}. How many diagonals do the following polygons have?
a) pentagon
b) hexagon
c) heptagon
d) octagon
{{exercise_number}}. Are there any polygons with the same number of diagonals as sides?
{{exercise_number}}. Find polygons that have
a) less diagonals than sides;
b) more diagonals than sides;
c) the same number of diagonals as sides.
{{exercise_number}}. How many times do the sides of a triangle intersect
a) a line segment;
b) the sides of a rectangle;
c) the sides of a convex quadrilateral;
d) the sides of a quadrilateral?
{{exercise_number}}. Draw a quadrilateral which can be divided into
a) three polygons with one line;
b) three triangles with one line.
{{exercise_number}}. How many rectangles are there in the images? (→)
a)
b)
c)
Quiz
How can you plant \latex{ 10 } trees in five rows so there are \latex{ 4 } trees in each row?
