Question

Известны координаты вершин треугольника АВС: А (1;3), В (2;0), С (6;-2). Найдите длину медианы АМ.

Answer 1

Точка М - середина отрезка ВС.
Найдем координаты точки М (для этого сложим координаты точек В и С, между которыми лежит точка М):
$( \frac{2+6}{2}, \frac{0-2}{2} )=(4,-1)$
Теперь найдем координаты вектора АМ:
$AM=(1-4, 3-(-1))=(-3, 4)$
Найдем длину AM:
$|AM|= \sqrt{(-3)^2+4^2}= \sqrt{9+16} = \sqrt{25} =5$
Ответ: 5.

Answer 2

Finding the Length of Median AM in Triangle ABC

To find the length of the median AM in 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) * √(2b^2 + 2c^2 - a^2)

Where a, b, and c are the lengths of the sides of the triangle. In this case, we can use the coordinates of points A, B, and C to calculate the lengths of the sides of the triangle.

First, let's calculate the lengths of the sides of triangle ABC.

The length of side AB: AB = √((x2 - x1)^2 + (y2 - y1)^2) = √((2 - 1)^2 + (0 - 3)^2) = √(1^2 + (-3)^2) = √(1 + 9) = √10

The length of side BC: BC = √((x3 - x2)^2 + (y3 - y2)^2) = √((6 - 2)^2 + (-2 - 0)^2) = √(4^2 + (-2)^2) = √(16 + 4) = √20

The length of side CA: CA = √((x1 - x3)^2 + (y1 - y3)^2) = √((1 - 6)^2 + (3 - (-2))^2) = √((-5)^2 + (3 + 2)^2) = √(25 + 25) = √50

Now that we have the lengths of the sides of triangle ABC, we can proceed to calculate the length of median AM.

Using the formula: Length of Median AM = (1/2) * √(2b^2 + 2c^2 - a^2)

Substitute the values: Length of Median AM = (1/2) * √(2(20) + 2(50) - 10) Length of Median AM = (1/2) * √(40 + 100 - 10) Length of Median AM = (1/2) * √(130) Length of Median AM = 0.5 * √130 Length of Median AM ≈ 6.4

So, the length of median AM in triangle ABC is approximately 6.4.