There are points in the coordinate system A (-1; 0), B (5; 0), C (3; 2).

Find such point D that the polygonpolygonpolygon ABCD is an isosceles trapezoidisosceles trapezoidisosceles trapezoid. Calculate lengths of bases of this trapezoid.

There is the parallelogramparallelogramparallelogram ABCD in the coordinate systemcoordinate systemcoordinate system.

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Na rysunku na układzie współrzędnych znajduje się równoległobok o wierzchołkach A (0, 5) B (5, 5) C (3, -1) D (-2, -1).

a) Are abscissas of points C and D the same?

b) How many points with both positive coordinates are inside the parallelogramparallelogramparallelogram ABCD?

c) Give the difference between the ordinate and the abscissa of the point B.

d) Calculate the length of the shorter side of the parallelogramparallelogramparallelogram ABCD.

There are points A (3; 2) and B (-3; 2).

Find such points C and D that the centre of symmetry of the rectanglerectanglerectangle ABCD is the beginning of the coordinate systemcoordinate systemcoordinate system. Calculate lengths of sides of this rectanglerectanglerectangle.

Mark points in the coordinate systemcoordinate systemcoordinate system A (-1; -1), B ( 2; 2), C (0; 4).

Give coordinates of such point D that the obtained tetragon is:

a) a rectanglerectanglerectangle that is not a square,
b) a square,
c) a right‑angled trapezoid that is not a rectanglerectanglerectangle,
d) a deltoiddeltoiddeltoid.

Is there a solution for each case?

An extra task:

Calculate the length of the longest side of the ABC triangletriangletriangle whose vertices are A (-3; -1), B (2; -1), C (2; 4).

There is the ABC triangle whose vertices are A (1; 2), B (3; 5), C (-1; 2). Which of the points is located inside the ABC triangle?

1. G (4; 1)

2. H (2; 1)

3. I (3; 5)

4. J (-1; 3)

Mark the isosceles trapezoid ABCD in the coordinate system. Its longer base AB’s length is 8 and shorter CD is 4. Mark it in such a way that the Y axis is the axis of symmetry of this trapezoid. Describe the solution in English.