Question

ПОМОГИТЕ ПОЖАЛУЙСТА!! дам 40 баллов!! Четырехугольник АВСD задан координатами своих вершин A(-1;1) , B(3;3), C(2;-2), D(-2;-1). Найдите синус угла между его диагоналями.

Answer 1

Ответ:

Вектора

AC (3;-3)  Длина 3√2

BD (-5;-4) Длина √(5^2+4^2)=√41

Косинус искомого угла

| AC * BD | / | AC | / | BD | = | 3*(-5)+3*4|  / 3√2 / √41 = 1 / √82

Синус

√ ( 1 - 1/82) = 9 / √ 82

Объяснение:

Answer 2

Finding the Diagonals of Quadrilateral ABCD

To find the diagonals of quadrilateral ABCD, we need to determine the lengths of the line segments connecting the opposite vertices. The diagonals of ABCD are AC and BD.

The coordinates of the given vertices are: A(-1, 1) B(3, 3) C(2, -2) D(-2, -1)

Using the distance formula, we can calculate the lengths of the diagonals AC and BD.

The distance formula is given by: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the lengths of the diagonals:

For diagonal AC: AC = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((2 - (-1))^2 + (-2 - 1)^2) = sqrt((3)^2 + (-3)^2) = sqrt(9 + 9) = sqrt(18)

For diagonal BD: BD = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-2 - 3)^2 + (-1 - 3)^2) = sqrt((-5)^2 + (-4)^2) = sqrt(25 + 16) = sqrt(41)

Now that we have the lengths of the diagonals AC and BD, we can proceed to find the sine of the angle between them.

Finding the Sine of the Angle between the Diagonals

To find the sine of the angle between the diagonals, we can use the formula:

sin(theta) = (2 * Area of Quadrilateral ABCD) / (AC * BD)

The area of a quadrilateral can be calculated using the Shoelace formula or by dividing it into two triangles and finding their areas. Since we already have the coordinates of the vertices, we can use the Shoelace formula.

The Shoelace formula is given by: Area = 0.5 * |(x1 * y2 + x2 * y3 + x3 * y1) - (y1 * x2 + y2 * x3 + y3 * x1)|

Let's calculate the area of quadrilateral ABCD:

Area = 0.5 * |(-1 * 3 + 3 * (-2) + 2 * 1) - (1 * 3 + 3 * 2 + (-2) * (-1))| = 0.5 * |(-3 - 6 + 2) - (3 + 6 + 2)| = 0.5 * |-7 - 11| = 0.5 * |-18| = 9

Now, we can calculate the sine of the angle between the diagonals:

sin(theta) = (2 * Area of Quadrilateral ABCD) / (AC * BD) = (2 * 9) / (sqrt(18) * sqrt(41)) = 18 / (sqrt(18) * sqrt(41)) = 18 / (sqrt(18 * 41)) = 18 / (sqrt(738)) ≈ 0.613

Therefore, the sine of the angle between the diagonals of quadrilateral ABCD is approximately 0.613.

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