Question

Дано А(-6;1),В(0;5),С(6;-4),Д(0;-8) Докажите , что АВСД -параллелограмм , и найдите его периметр

Answer 1

Найдем длины сторон четырехугольника АВСD.
АВ=√36+16=√52=2√13,
ВС=√36+81=√117=3√13,
СD=√36+16=√52=2√13,
АD=√36+81=√117=3√13. АВ=СD. ВС=АD.
Найдем периметр параллелограмма АВСD (противоположные стороны равны) Р(АВСD)=АВ+ВС+СD+АD=10√13
Ответ : 10√13.

Answer 2

Proof that ABCD is a Parallelogram

To prove that ABCD is a parallelogram, we need to show that opposite sides are parallel. Let's examine the coordinates of the given points A(-6, 1), B(0, 5), C(6, -4), and D(0, -8).

The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:

slope = (y2 - y1) / (x2 - x1)

Let's calculate the slopes of the lines AB, BC, CD, and DA.

Slope of AB: (x1, y1) = (-6, 1) (x2, y2) = (0, 5)

slope_AB = (5 - 1) / (0 - (-6)) = 4 / 6 = 2/3

Slope of BC: (x1, y1) = (0, 5) (x2, y2) = (6, -4)

slope_BC = (-4 - 5) / (6 - 0) = -9 / 6 = -3/2

Slope of CD: (x1, y1) = (6, -4) (x2, y2) = (0, -8)

slope_CD = (-8 - (-4)) / (0 - 6) = -4 / (-6) = 2/3

Slope of DA: (x1, y1) = (0, -8) (x2, y2) = (-6, 1)

slope_DA = (1 - (-8)) / (-6 - 0) = 9 / (-6) = -3/2

We can see that the slopes of AB and CD are equal (2/3), and the slopes of BC and DA are equal (-3/2). Therefore, opposite sides AB and CD are parallel, and opposite sides BC and DA are parallel.

Hence, we have proved that ABCD is a parallelogram.

Perimeter of ABCD

To find the perimeter of ABCD, we need to calculate the lengths of its sides.

Length of AB: (x1, y1) = (-6, 1) (x2, y2) = (0, 5)

AB = sqrt((0 - (-6))^2 + (5 - 1)^2) = sqrt(6^2 + 4^2) = sqrt(36 + 16) = sqrt(52)

Length of BC: (x1, y1) = (0, 5) (x2, y2) = (6, -4)

BC = sqrt((6 - 0)^2 + (-4 - 5)^2) = sqrt(6^2 + (-9)^2) = sqrt(36 + 81) = sqrt(117)

Length of CD: (x1, y1) = (6, -4) (x2, y2) = (0, -8)

CD = sqrt((0 - 6)^2 + (-8 - (-4))^2) = sqrt((-6)^2 + (-4)^2) = sqrt(36 + 16) = sqrt(52)

Length of DA: (x1, y1) = (0, -8) (x2, y2) = (-6, 1)

DA = sqrt((-6 - 0)^2 + (1 - (-8))^2) = sqrt((-6)^2 + 9^2) = sqrt(36 + 81) = sqrt(117)

Now, we can calculate the perimeter of ABCD by summing the lengths of its sides:

Perimeter = AB + BC + CD + DA = sqrt(52) + sqrt(117) + sqrt(52) + sqrt(117)

Therefore, the perimeter of ABCD is 2(sqrt(52) + sqrt(117)).