Question

Точка М равноудалена от всех вершин прямоугольного треугольника АВС ( угол С=90 грудусов), АС = ВС=4см. Расстояние от точки М до плоскости АВС равно 2 корня из 3 см.Доказать что (АМВ) перпендикулярна (АВС).Найти1) Угол между (ВМС) и (АВС)2) Угол между (МС) и (АВС)3) Объем пирамиды МАВС4) Площадь полной поверхности МАВСПОМОГИТЕ ПОЖАЛУЙСТА !!))) Очень надо, ни то 2 поставят)Нужен рисунок)Заранее благодарю))

Answer 1

Ответ будет в приложенном рисунке.

Answer 2

Given Information

We are given the following information: - Point M is equidistant from all vertices of right triangle ABC, where angle C is 90 degrees. - AC = BC = 4 cm. - The distance from point M to the plane ABC is 2√3 cm.

Proof that (AMV) is perpendicular to (ABC)

To prove that line segment AMV is perpendicular to triangle ABC, we need to show that the slopes of AM and MV are negative reciprocals of each other.

Let's assume the coordinates of point A are (0, 0), and the coordinates of point C are (4, 0). Since AC = BC, the coordinates of point B will be (0, 4).

Let's assume the coordinates of point M are (x, y). Since point M is equidistant from all vertices of triangle ABC, the distance from point M to point A will be equal to the distance from point M to point B.

Using the distance formula, we can set up the following equation: √((x - 0)^2 + (y - 0)^2) = √((x - 0)^2 + (y - 4)^2)

Simplifying the equation, we get: x^2 + y^2 = x^2 + (y - 4)^2

Expanding and simplifying further, we get: y^2 = (y - 4)^2

Solving this equation, we find that y = 2.

Therefore, the coordinates of point M are (x, 2).

Now, let's find the slope of line segment AM. The slope of a line passing through points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).

The slope of line segment AM is: (2 - 0) / (x - 0) = 2 / x

Next, let's find the slope of line segment MV. The slope of a line passing through points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).

The coordinates of point V are (4, 2). Therefore, the slope of line segment MV is: (2 - 2) / (4 - x) = 0 / (4 - x) = 0

Since the slope of line segment MV is 0, the slope of line segment AM is undefined (since it is a vertical line).

Therefore, the slopes of AM and MV are negative reciprocals of each other, which means that line segment AMV is perpendicular to line segment ABC.

Finding the Angle between (BMC) and (ABC)

To find the angle between line segment BMC and triangle ABC, we need to find the angle between line segment BM and line segment BC.

Since line segment BM is perpendicular to line segment BC (as proved earlier), the angle between line segment BMC and triangle ABC is 90 degrees.

Finding the Angle between (MC) and (ABC)

To find the angle between line segment MC and triangle ABC, we need to find the angle between line segment MC and line segment BC.

Since line segment MC is perpendicular to line segment BC (as proved earlier), the angle between line segment MC and triangle ABC is 90 degrees.

Finding the Volume of Pyramid MAVC

To find the volume of the pyramid MAVC, we can use the formula for the volume of a pyramid: V = (1/3) * base area * height.

The base of the pyramid is triangle ABC, and the height is the distance from point M to the plane ABC.

The area of triangle ABC can be found using the formula for the area of a triangle: A = (1/2) * base * height.

The base of triangle ABC is BC, which is 4 cm, and the height is AC, which is also 4 cm.

Therefore, the area of triangle ABC is: A = (1/2) * 4 cm * 4 cm = 8 cm^2

The height of the pyramid is given as 2√3 cm.

Using the formula for the volume of a pyramid, we can calculate the volume of pyramid MAVC: V = (1/3) * 8 cm^2 * 2√3 cm = (16/3)√3 cm^3

Finding the Surface Area of Pyramid MAVC

To find the surface area of the pyramid MAVC, we need to find the area of the base (triangle ABC) and the area of the lateral faces.

The area of triangle ABC is 8 cm^2 (as calculated earlier).

The area of each lateral face can be found using the formula for the area of a triangle: A = (1/2) * base * height.

The base of each lateral face is BC, which is 4 cm, and the height is the distance from point M to the plane ABC, which is given as 2√3 cm.

Therefore, the area of each lateral face is: A = (1/2) * 4 cm * 2√3 cm = 4√3 cm^2

Since the pyramid has four lateral faces, the total area of the lateral faces is 4 times the area of each lateral face, which is 4 * 4√3 cm^2 = 16√3 cm^2.

The total surface area of the pyramid MAVC is the sum of the area of the base and the area of the lateral faces: Surface Area = 8 cm^2 + 16√3 cm^2 = 8 cm^2 + 16√3 cm^2

Conclusion

In conclusion: 1) The angle between line segment BMC and triangle ABC is 90 degrees. 2) The angle between line segment MC and triangle ABC is 90 degrees. 3) The volume of pyramid MAVC is (16/3)√3 cm^3. 4) The surface area of pyramid MAVC is 8 cm^2 + 16√3 cm^2.

Please note that a diagram is not provided in the search results, so I am unable to provide a visual representation. However, I hope the explanations provided above are helpful.