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Mixed exercises
{{exercise_number}}. For which quadrilaterals are the following statements true? Answer with their numbers.
\latex{ 1. }
\latex{ 2. }
\latex{ 3. }
\latex{ 4. }
\latex{ 5. }
\latex{ 6. }
\latex{ 7. }
\latex{ 8. }
\latex{ 9. }
- Have exactly two axes of symmetry.
- Have an axis of symmetry passing through a vertex.
- Have an odd number of axes of symmetry.
- Do not have an axis of symmetry passing through a vertex.
- Do not have an axis of symmetry.
- Regular quadrilaterals.
{{exercise_number}}. Determine the rule. What is the next plane figure of the sequence?
\latex{ a }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
?
{{exercise_number}}. Decide whether the following statements are true or false.
- There is an isosceles triangle that has exactly two axes of symmetry.
- Every isosceles triangle has at least two equal angles.
- If a quadrilateral is axially symmetric, its sides are equal.
- Every axially symmetric quadrilateral is a kite.
- If the diagonals of a quadrilateral are equal and perpendicular to each other, the quadrilateral is a kite.
- There is an axially symmetric quadrilateral whose diagonal is not an axis of symmetry.
{{exercise_number}}. Construct a square with \latex{ 6\, cm } long diagonals.
{{exercise_number}}. There is a line \latex{ l } and a point \latex{ P } outside it. Construct a square with one of its vertices being point \latex{ P } and one of its diagonals being on line \latex{ l }.
{{exercise_number}}. Construct a rectangle with one of its sides being \latex{ 4 } \latex{ centimetres } longer than the other and its perimeter being \latex{ 24\, cm }.
{{exercise_number}}. Construct a rectangle with \latex{ 2 } and \latex{ 4\, cm } long sides.
- Reflect it across the line of one of its diagonals.
- What type of polygon is formed by the original rectangle and its mirror image?
{{exercise_number}}. Construct an isosceles triangle with a perimeter of \latex{ 17\, cm } and
- a base of \latex{ 5\, cm };
- \latex{ 3\, cm } long sides.
{{exercise_number}}. Construct an arbitrary \latex{ ABC } triangle. Construct axis \latex{l} and point \latex{ C } if you know that \latex{ A' = B } when triangle \latex{ ABC } is reflected across axis \latex{l}.
{{exercise_number}}. Construct a triangle with \latex{ 5;\, 6 } and \latex{ 9\, cm } long sides.
- Construct the points found at the same distance from the endpoints of the \latex{ 5\, cm } long side.
- Construct the points found at the same distance from the endpoints of the \latex{ 6\, cm } long side.
- Where is the point found at the same distance from all three vertices of the triangle?
- Construct a circle passing through all three vertices of the triangle.
{{exercise_number}}. Construct a triangle congruent to the triangle in the image. Then, construct the bisector of angle \latex{\alpha }. Reflect the triangle across the resulting angle bisector. What type of polygon is formed by the original triangle and its mirror image?
\latex{\alpha}
\latex{a}
\latex{b}
\latex{c}
{{exercise_number}}. There is a triangle \latex{ ABC } with a point \latex{ P } inside it. Reflect point \latex{ P } across the sides of the triangle and connect the reflected points with the endpoints of the side used as the mirror line.
What type of polygon is the resulting figure? How many times larger is the area of the resulting polygon than that of the \latex{ ABC } triangle?
\latex{C}
\latex{P}
\latex{A}
\latex{B}
{{exercise_number}}. Construct a kite with \latex{ 4\, cm } and \latex{ 5\, cm } long sides that form a \latex{ 120° } angle.
{{exercise_number}}. Construct a kite with \latex{ 4\, cm } long sides that form a \latex{ 135° } angle. The other two sides are \latex{ 5\, cm } long.
{{exercise_number}}. Construct a kite if the lengths of two different sides are \latex{ 4 } and \latex{ 5\, cm } and one of the angles is a right angle. How many kites can be constructed?
{{exercise_number}}. Construct an isosceles trapezium with a \latex{ 7\, cm } long base, \latex{ 3\, cm } long sides and \latex{ 45° } base angles. Construct the perpendicular bisectors of the sides. What do you notice?
{{exercise_number}}. Construct a rhombus with a \latex{ 135° } angle and \latex{ 4\, cm } long sides.
{{exercise_number}}. Construct a polygon with \latex{ 6 } axes of symmetry.
{{exercise_number}}. What polygons with less than \latex{ 10 } sides can be divided into rhombuses?
{{exercise_number}}. Calculate how far Andie's house is from the bus stop using the image.
\latex{ 500\; m }
\latex{ 30 ° }
