Question

E(4:12) F(-4;-10) G (-2;6) H (4;-2) a) Найти координатв векторов EF GH б) Длину вектора FG в) координаты точки О-середины EF,координаты точки W-середины Gh д) написать уравнение окружности с диаметром FG е) написать уравнение прямой FH

Answer 1

В д) нужен центр , а в е) Не знаю

Answer 2

a) Найти координаты векторов EF и GH

To find the coordinates of vectors EF and GH, we subtract the coordinates of the initial point from the coordinates of the terminal point.

For vector EF: EF = (x2 - x1, y2 - y1) = (-4 - 4, -10 - 12) = (-8, -22)

For vector GH: GH = (x2 - x1, y2 - y1) = (-2 - 4, 6 - (-2)) = (-6, 8)

Therefore, the coordinates of vector EF are (-8, -22) and the coordinates of vector GH are (-6, 8).

b) Длина вектора FG

To find the length of vector FG, we can use the distance formula, which is the square root of the sum of the squares of the differences in the coordinates.

FG = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-4 - (-2))^2 + (-10 - 6)^2) = sqrt((-2)^2 + (-16)^2) = sqrt(4 + 256) = sqrt(260) ≈ 16.125

Therefore, the length of vector FG is approximately 16.125.

в) Координаты точки O - середины EF, координаты точки W - середины GH

To find the coordinates of the midpoint of a line segment, we average the x-coordinates and the y-coordinates of the endpoints.

For the midpoint of EF (point O): x-coordinate of O = (x1 + x2) / 2 = (4 + (-4)) / 2 = 0 / 2 = 0 y-coordinate of O = (y1 + y2) / 2 = (12 + (-10)) / 2 = 2 / 2 = 1

Therefore, the coordinates of point O are (0, 1).

For the midpoint of GH (point W): x-coordinate of W = (x1 + x2) / 2 = (-2 + 4) / 2 = 2 / 2 = 1 y-coordinate of W = (y1 + y2) / 2 = (6 + (-2)) / 2 = 4 / 2 = 2

Therefore, the coordinates of point W are (1, 2).

д) Написать уравнение окружности с диаметром FG

To write the equation of a circle with diameter FG, we need to find the center of the circle and the radius.

The center of the circle is the midpoint of FG, which we already found to be (1, 2).

The radius of the circle is half of the length of FG, which is 16.125 / 2 = 8.0625.

Therefore, the equation of the circle is:

(x - 1)^2 + (y - 2)^2 = 8.0625^2

е) Написать уравнение прямой FH

To write the equation of the line FH, we need to find the slope of the line and a point that lies on the line.

The slope of the line can be found using the formula:

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

For points F(-4, -10) and H(4, -2), the slope is:

slope = (-2 - (-10)) / (4 - (-4)) = 8 / 8 = 1

Now we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

Using point F(-4, -10):

y - (-10) = 1(x - (-4))

Simplifying:

y + 10 = x + 4

Rearranging the equation:

y = x - 6

Therefore, the equation of the line FH is y = x - 6.