Question

Стороны правильного треугольника АВС равны 3 корня из 3. Найдите длину вектра АМ, где М середина ВС. Подробное решение пожалуйста.

Answer 1

Если М - середина ВС, получается, что СМ=МВ. Так как треугольник АВС равносторонний, то АМ явлеятся в треугольнике высотой, медианой и биссектрисой.

Рассмотрим треугольник АВМ:

1. АВ = 3 корня из 3

2. ВМ = 3 корня из 3 / 2 (так как М - середина ВС)

3. АВС - прямоугольный, так как АМ - высота в равностороннем треугольнике

По теореме Пифагора :

АВ в квадрате = ВМ в квадрате + АМ в квадрате

(3 корня из 3)^2 = (3 корня из 3/2)^2 + АМ ^2 

Отсюда АМ^2 = 27 - 27/4 

4АМ^2 = 108 - 27

4АМ^2 = 81

АМ^2 = 81/4

АМ = 9/2 = 4.5

Answer 2

Problem Statement

We are given that the sides of a right triangle ABC are equal to 3 times the square root of 3. We need to find the length of vector AM, where M is the midpoint of side BC.

Solution

To solve this problem, we can use the concept of vectors and the properties of right triangles.

Let's start by finding the length of side BC. We are given that the sides of the right triangle ABC are equal to 3 times the square root of 3. Therefore, the length of side BC is 3√3.

Next, let's find the length of side AC. Since ABC is a right triangle, we can use the Pythagorean theorem to find the length of AC. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (AC) is equal to the sum of the squares of the lengths of the other two sides (AB and BC).

Using the Pythagorean theorem, we have:

AC^2 = AB^2 + BC^2 AC^2 = (3√3)^2 + (3√3)^2 AC^2 = 9*3 + 9*3 AC^2 = 27 + 27 AC^2 = 54

Taking the square root of both sides, we get:

AC = √54 AC = 3√6

Now, let's find the length of vector AM. Since M is the midpoint of side BC, we can use the midpoint formula to find the coordinates of point M. The midpoint formula states that the coordinates of the midpoint (M) of a line segment with endpoints (x1, y1) and (x2, y2) are given by:

M = ((x1 + x2)/2, (y1 + y2)/2)

In this case, the coordinates of point B are (0, 0) and the coordinates of point C are (3√3, 0). Therefore, the coordinates of point M are:

M = ((0 + 3√3)/2, (0 + 0)/2) M = (3√3/2, 0)

Now, let's find the length of vector AM. The length of a vector is given by the formula:

|AM| = √(x^2 + y^2)

In this case, the coordinates of point A are (0, 3√6) and the coordinates of point M are (3√3/2, 0). Therefore, the length of vector AM is:

|AM| = √((3√3/2 - 0)^2 + (0 - 3√6)^2) |AM| = √((9/4)*3 + 9*6) |AM| = √(27/4 + 54) |AM| = √(27 + 216)/4 |AM| = √243/4 |AM| = 3√3/2

Therefore, the length of vector AM is 3√3/2.

Answer

The length of vector AM, where M is the midpoint of side BC in the right triangle ABC with sides equal to 3 times the square root of 3, is 3√3/2.