Доказать, что ABCD - ромб, если A(7;4), B(11;2), C(7;0), D(3;2). Найти площадь ромба ABCD.
To prove that ABCD is a rhombus, we first need to show that all four sides are equal in length and that the opposite angles are congruent.To begin, we will use the distance formula to find the lengths of each side.
Side AB = sqrt((11-7)^2 + (2-4)^2) = sqrt(16 + 4) = sqrt(20)
Side BC = sqrt((7-11)^2 + (0-2)^2) = sqrt(16 + 4) = sqrt(20)
Side CD = sqrt((3-7)^2 + (2-0)^2) = sqrt(16 + 4) = sqrt(20)
Side DA = sqrt((3-7)^2 + (2-4)^2) = sqrt(16 + 4) = sqrt(20)
As we can see, all four sides have the same length, which proves that ABCD is a rhombus.
Next, we can use the slope formula to find the slopes of each side.
Slope AB = (2-4)/(11-7) = -0.5
Slope BC = (0-2)/(7-11) = -0.5
Slope CD = (2-0)/(3-7) = -0.5
Slope DA = (2-4)/(3-7) = -0.5
Since all four sides have the same slope, we can conclude that all four angles are congruent and ABCD is a rhombus.
Now, to find the area of the rhombus, we can use the formula A = (1/2) * d1 *d2, where d1 and d2 are the lengths of the diagonals.
Diagonal AC = sqrt((7-7)^2 + (4-0)^2) = sqrt(16) = 4
Diagonal BD = sqrt((11-3)^2 + (2-2)^2) = sqrt(64) = 8
Therefore, the area of ABCD = (1/2) * 4 * 8 = 16 square units.
Hence, we have proven that ABCD is a rhombus and have found its area to be 16 square units.
Disclaimer: Always double check your work and make sure to cite any outside sources used in your proof.
