Question

Знайдіть довжину медіани AM трикутника ABC якщо А( 1 0 2 ). В(5 4 -3) С(-3 -2 7 )

Answer 1

Ответ:  1

Объяснение:

М- середина ВС, сумма соответствующих координат делится на два. так ищем середину, затем, имея два конца медианы - А и М,  от координат точки М отнимаем соответствующие координаты точки А, возводим разности в квадрат и суммируем результаты. Извлекаем корень квадратный из этой суммы, получаем длину медианы.

М(х;у;z)

x=(5+(-3))/2=1

y=(4+(-2))/2=1

z=(-3+7)/2=2

М(1:1;2)

АМ=√(1-1)²+(1-0)²+(2-2)²=1

Answer 2

Finding the Length of Median AM of Triangle ABC

To find the length of median AM of triangle ABC, we can use the formula for the length of a median in a triangle, which is given by:

Length of Median AM = 1/2 * sqrt(2 * (b^2 + c^2) - a^2)

Where: - a, b, and c are the lengths of the sides of the triangle.

Given the coordinates of points A, B, and C as: A(1, 0, 2), B(5, 4, -3), C(-3, -2, 7)

We can calculate the lengths of the sides of the triangle using the distance formula, and then use the formula for the length of the median to find the length of median AM.

Calculating the Lengths of the Sides of Triangle ABC

Using the distance formula, the lengths of the sides of triangle ABC can be calculated as follows:

Length of side AB: ``` AB = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2) = sqrt((5 - 1)^2 + (4 - 0)^2 + (-3 - 2)^2) = sqrt(16 + 16 + 25) = sqrt(57) ```

Length of side AC: ``` AC = sqrt((x3 - x1)^2 + (y3 - y1)^2 + (z3 - z1)^2) = sqrt((-3 - 1)^2 + (-2 - 0)^2 + (7 - 2)^2) = sqrt(16 + 4 + 25) = sqrt(45) ```

Length of side BC: ``` BC = sqrt((x3 - x2)^2 + (y3 - y2)^2 + (z3 - z2)^2) = sqrt((-3 - 5)^2 + (-2 - 4)^2 + (7 + 3)^2) = sqrt(64 + 36 + 100) = sqrt(200) = 10*sqrt(2) ```

Calculating the Length of Median AM

Now, we can use the formula for the length of median AM to find its length:

Length of Median AM = 1/2 * sqrt(2 * (AB^2 + AC^2) - BC^2) ``` Length of Median AM = 1/2 * sqrt(2 * (57 + 45) - (10*sqrt(2))^2) = 1/2 * sqrt(2 * 102 - 200) = 1/2 * sqrt(204 - 200) = 1/2 * sqrt(4) = 1 ```

Conclusion

The length of median AM of triangle ABC is 1.