Question

Даны вершины четырехугольника A(-4; -2), B(-3;1),C(4;3), D(5: -3) Показать, что середины сторон этого четырехугольника являются вершинами параллелограммаПомогите решить, пожалуйста!

Answer 1

Смотри, сначала по формуле ищешь сами точки :
sqrt(x*x+y*y)
А дальше доказываешь, что точки А и В расположены друг от друга на одинаковом расстоянии, как и точки С и D
Тоже самое насчет точек В и С и А и D

Answer 2

Problem Analysis

To show that the midpoints of the sides of a quadrilateral form the vertices of a parallelogram, we need to find the midpoints of the sides of the given quadrilateral and then show that the opposite sides of the resulting parallelogram are parallel.

Solution

Let's find the midpoints of the sides of the quadrilateral ABCD:

1. Midpoint of AB: - The coordinates of A are (-4, -2) and the coordinates of B are (-3, 1). - To find the midpoint, we average the x-coordinates and the y-coordinates separately. - The x-coordinate of the midpoint is (-4 + -3) / 2 = -7/2 = -3.5. - The y-coordinate of the midpoint is (-2 + 1) / 2 = -1/2 = -0.5. - Therefore, the midpoint of AB is M1(-3.5, -0.5).

2. Midpoint of BC: - The coordinates of B are (-3, 1) and the coordinates of C are (4, 3). - Using the same process as above, we find that the midpoint of BC is M2(0.5, 2).

3. Midpoint of CD: - The coordinates of C are (4, 3) and the coordinates of D are (5, -3). - Using the same process as above, we find that the midpoint of CD is M3(4.5, 0).

4. Midpoint of DA: - The coordinates of D are (5, -3) and the coordinates of A are (-4, -2). - Using the same process as above, we find that the midpoint of DA is M4(0.5, -2.5).

Now, let's show that the opposite sides of the resulting parallelogram are parallel:

1. Opposite sides M1M3 and M2M4: - The slope of M1M3 is (0 - (-0.5)) / (4.5 - (-3.5)) = -0.5 / 8 = -1/16. - The slope of M2M4 is (-2.5 - 2) / (0.5 - 0.5) = -4.5 / 0 = undefined. - Since the slopes are different, the sides M1M3 and M2M4 are not parallel.

2. Opposite sides M2M4 and M3M1: - The slope of M2M4 is (-2.5 - 2) / (0.5 - 0.5) = -4.5 / 0 = undefined. - The slope of M3M1 is (-0.5 - (-0.5)) / (-3.5 - 4.5) = 0 / (-8) = 0. - Since the slopes are different, the sides M2M4 and M3M1 are not parallel.

Since the opposite sides of the resulting parallelogram are not parallel, we can conclude that the midpoints of the sides of the given quadrilateral ABCD do not form the vertices of a parallelogram.

Conclusion

The midpoints of the sides of the given quadrilateral ABCD do not form the vertices of a parallelogram.