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The properties of rectangles
Polygons with four sides are called quadrilaterals.
Terminology related to quadrilaterals:
– adjacent vertices (\latex{ B } and \latex{ C } or \latex{ A } and \latex{ D });
– opposite vertices (\latex{ A } and \latex{ C } or \latex{ B } and \latex{ D });
– adjacent sides (\latex{\large a } and \latex{\large b } or \latex{\large a } and \latex{\large d });
– opposite sides (\latex{\large a } and \latex{\large c } or \latex{\large b } and \latex{\large d });
– diagonals (\latex{\large e } = \latex{ A }\latex{ C } and \latex{\large f } = \latex{ B }\latex{ D }).
– opposite vertices (\latex{ A } and \latex{ C } or \latex{ B } and \latex{ D });
– adjacent sides (\latex{\large a } and \latex{\large b } or \latex{\large a } and \latex{\large d });
– opposite sides (\latex{\large a } and \latex{\large c } or \latex{\large b } and \latex{\large d });
– diagonals (\latex{\large e } = \latex{ A }\latex{ C } and \latex{\large f } = \latex{ B }\latex{ D }).
\latex{ A }
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ a }
\latex{ c }
\latex{ f }
\latex{ e }
\latex{ d }
\latex{ b }
The rectangle
A rectangle is a quadrilateral whose adjacent sides are perpendicular to each other.
\latex{ A }
\latex{ D }
\latex{ B }
\latex{ C }
Grab a rectangular piece of paper and fold it in the following way.
Fold point \latex{ A } to \latex{ D },
and \latex{ B } to \latex{ C }
and \latex{ B } to \latex{ C }
Fold point
\latex{ B }(\latex{ C }) to \latex{ A }(\latex{ D })
\latex{ B }(\latex{ C }) to \latex{ A }(\latex{ D })
Fold the paper
along line \latex{ C }\latex{ O }
along line \latex{ C }\latex{ O }
Unfold it
This is what
you get
you get
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ O }
\latex{ O }
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ O }
\latex{ e }
\latex{ f }
\latex{ C }
Properties of the rectangle:
– all angles are equal (\latex{ 90°});
– the opposite sides are parallel (\latex{ A }\latex{ B } \latex{\parallel} \latex{ D }\latex{ C } and \latex{ A }\latex{ D } \latex{\parallel} \latex{ B }\latex{ C });
– the opposite sides are equal in length (\latex{ A }\latex{ B } = \latex{ D }\latex{ C }= \latex{\large a } and \latex{ A }\latex{ D } = \latex{ B }\latex{ C } = \latex{\large b });
– the diagonals are equal in length (\latex{ A }\latex{ C } = \latex{ B }\latex{ D }; \latex{\large e } = \latex{ f });
– the diagonals bisect each other (\latex{ A }\latex{ O } = \latex{ C }\latex{ O } = \latex{ B }\latex{ O } = \latex{ D }\latex{ O });
– it can be folded in half in two ways so that the two halves overlap (along the red lines);
– the diagonals cut it into two congruent right-angled triangles.
– the opposite sides are parallel (\latex{ A }\latex{ B } \latex{\parallel} \latex{ D }\latex{ C } and \latex{ A }\latex{ D } \latex{\parallel} \latex{ B }\latex{ C });
– the opposite sides are equal in length (\latex{ A }\latex{ B } = \latex{ D }\latex{ C }= \latex{\large a } and \latex{ A }\latex{ D } = \latex{ B }\latex{ C } = \latex{\large b });
– the diagonals are equal in length (\latex{ A }\latex{ C } = \latex{ B }\latex{ D }; \latex{\large e } = \latex{ f });
– the diagonals bisect each other (\latex{ A }\latex{ O } = \latex{ C }\latex{ O } = \latex{ B }\latex{ O } = \latex{ D }\latex{ O });
– it can be folded in half in two ways so that the two halves overlap (along the red lines);
– the diagonals cut it into two congruent right-angled triangles.
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ a }
\latex{ e }
\latex{ f }
\latex{O}
The square
A square is a rectangle whose sides are equal in length.
\latex{ A }
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
Make a square out of a rectangular piece of paper.
- Fold vertex \latex{ C } onto side \latex{ F }\latex{ B }.
- Cut it at point \latex{ C }, parallel to side \latex{ E }\latex{ F }.
Fold the resulting square in the following way.
\latex{ E }
\latex{ F }
\latex{ C }
\latex{ B }
\latex{ C }
This is what
you get
you get
Fold it in half along
diagonal \latex{ A }\latex{ C }
diagonal \latex{ A }\latex{ C }
Fold vertex \latex{ A }
onto vertex \latex{ C }
onto vertex \latex{ C }
Unfold it
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ A }
\latex{ C }
\latex{ O }
\latex{ O }
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ O }
\latex{ e }
\latex{ f }
Fold it in
half again
half again
Properties of the square:
– all angles are equal (\latex{ 90°});
– all sides are equal;
– the diagonals are equal and bisect each other;
– the diagonals are perpendicular to each other (\latex{ A }\latex{ C } ⊥ \latex{ B }\latex{ D }; \latex{ \large e } ⊥ \latex{ f } );
– the diagonals divide it into four congruent triangles;
– it can be folded in half in four places so that the two halves overlap.
– all sides are equal;
– the diagonals are equal and bisect each other;
– the diagonals are perpendicular to each other (\latex{ A }\latex{ C } ⊥ \latex{ B }\latex{ D }; \latex{ \large e } ⊥ \latex{ f } );
– the diagonals divide it into four congruent triangles;
– it can be folded in half in four places so that the two halves overlap.
Quadrilaterals can be grouped into sets:
\latex{ D }
\latex{ C }
\latex{ A }
\latex{ B }
\latex{ f }
\latex{ e }
Quadrilaterals
Rectangles
Squares
In the image, you can see that
– not every quadrilateral is a rectangle;
– not every rectangle is a square.
– not every rectangle is a square.
Exercises
{{exercise_number}}. List the letters of the quadrilaterals that are
a) squares;
b) rectangles.
Make a Venn diagram and write the letters in the corresponding sets.
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ J }
\latex{ K }
\latex{ I }
\latex{ H }
\latex{ G }
{{exercise_number}}. Make four triangles like the one shown in the image. (→)
Use them to make a
a) square;
b) rectangle that is not a square.
Draw the compositions.
{{exercise_number}}. Divide a square into four congruent parts by folding it. Draw the folded shapes as well.
Find various solutions.
Find various solutions.
{{exercise_number}}. Can a square be divided into five congruent parts?
{{exercise_number}}. Divide a rectangle into four congruent parts by folding. Draw the folded shapes as well.
Find various solutions.
{{exercise_number}}. Two skewers are the diagonals of a quadrilateral. Put them together in such a way that
a) you get a rectangle;
b) you get a square;
c) the two skewers are perpendicular to each other, but they do not form a square.
{{exercise_number}}. Are the following statements true or false?
a) If the diagonals of a rectangle are equal in length, the rectangle is a square.
b) If the diagonals are perpendicular to each other, the rectangle is a square.
c) If the diagonals of a rectangle bisect each other, then it is a square.
d) If any two sides of a rectangle are equal, it is a square.
e) If the rectangle has two equal sides, it is a square.
f) If two adjacent sides of a rectangle are equal in length, it is a square.
b) If the diagonals are perpendicular to each other, the rectangle is a square.
c) If the diagonals of a rectangle bisect each other, then it is a square.
d) If any two sides of a rectangle are equal, it is a square.
e) If the rectangle has two equal sides, it is a square.
f) If two adjacent sides of a rectangle are equal in length, it is a square.
{{exercise_number}}. How long are the sides of the squares shown in the image if the sides of the two smallest squares are considered \latex{ 1 } unit? Draw the figure on quad paper.
{{exercise_number}}. Four children cut squares and rectangles out of paper and noticed the following:
Anna: If we place two congruent squares next to each other, we get a rectangle.
Ben: If we cut a rectangle parallel to one of its sides, we get two rectangles.
Carl: If two congruent rectangles are placed next to each other, we definitely get a rectangle.
Dora: If two rectangles are not congruent, it is impossible to place them next to each other in such a way that we get a rectangle.
Ben: If we cut a rectangle parallel to one of its sides, we get two rectangles.
Carl: If two congruent rectangles are placed next to each other, we definitely get a rectangle.
Dora: If two rectangles are not congruent, it is impossible to place them next to each other in such a way that we get a rectangle.
Who is right? Who is not? Make and draw the appropriate squares and rectangles to justify your answer.
{{exercise_number}}. How many ways can you choose four of the given points so that they are the vertices of a square ? (→)
a)
b)
{{exercise_number}}. Using quad paper, draw quadrilaterals that have exactly
a) one;
b) two;
c) three;
d) four right angles.
{{exercise_number}}. How many congruent rectangles are in the image? (→)
a) Remove two matchsticks so that three squares remain.
b) Remove two matchsticks so that two squares remain.
b) Remove two matchsticks so that two squares remain.
Matchsticks cannot be placed on top of each other or stick out from the figure.
{{exercise_number}}. Based on your knowledge, list sports where the playing field is rectangular and sports where it is not.
{{exercise_number}}. Divide a square into
a) \latex{ 4 };
b) \latex{ 7 };
c) \latex{ 9 };
d) \latex{ 12 };
e) \latex{ 17 } smaller
(not necessarily congruent) squares.
Quiz
Congruent quadrilaterals were placed on the table one after the other. After placing down the eighth figure, the figure shown in the image was created on the table. In what order were the quadrilaterals placed on the table?
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ G }
\latex{ H }
\latex{ A }
