Question

Точки А(4; -1), В (2; 4), С (0; -1) являются вершинами параллелограмма АВСД. а) Найдите координаты вершины Д. б) Докажите, что параллелограмм АВСД является ромбом.

Answer 1

Finding the coordinates of point D

To find the coordinates of point D, we can use the fact that the opposite sides of a parallelogram are parallel and have the same length. We can start by finding the vector representing the line segment AB and then use it to find point D.

The vector representing the line segment AB can be found by subtracting the coordinates of point A from the coordinates of point B:

AB = (2 - 4, 4 - (-1)) = (-2, 5)

To find point D, we can add the vector AB to the coordinates of point C:

D = C + AB = (0, -1) + (-2, 5) = (-2, 4)

Therefore, the coordinates of point D are (-2, 4).

Proving that ABCD is a rhombus

To prove that ABCD is a rhombus, we need to show that all four sides of the parallelogram are congruent.

We can start by finding the lengths of the sides of the parallelogram. The length of a line segment can be found using the distance formula:

AB = sqrt((2 - 4)^2 + (4 - (-1))^2) = sqrt(4 + 25) = sqrt(29)

BC = sqrt((0 - 2)^2 + (-1 - 4)^2) = sqrt(4 + 25) = sqrt(29)

CD = sqrt((-2 - 0)^2 + (4 - (-1))^2) = sqrt(4 + 25) = sqrt(29)

DA = sqrt((4 - (-2))^2 + (-1 - 4)^2) = sqrt(36 + 25) = sqrt(61)

Since all four sides of the parallelogram have the same length, namely sqrt(29), we can conclude that ABCD is a rhombus. In summary: - The coordinates of point D are (-2, 4). - The parallelogram ABCD is a rhombus because all four sides have the same length, sqrt(29).