Question

100 Баллов, помогите пожалуйста, 1 задача, 10 класс, начало стереометрии. В кубе ABCD;A1B1C1D1, точка М на ребре А1В1, приэтом МВ=1\3А1В1. Постройте точку N пересечения прямой АМ на площади (ВВ1СС1). И найдите длину отрезка MN, если ребро куба равно 12 сантиметров.

Answer 1

Прямая АМ лежит в плоскости АА1В1В, которая пересекается с плоскостью ВВ1С1С по прямой ВВ1.
Поэтому надо продлить отрезок АМ до пересечения с продолжением ВВ1, где и получим точку N.
Находим B1N из пропорции для подобных треугольников:
 х/4 = 12/(12-4),
 х/4 = 12/8,
 2х = 12,
х = 12/2 = 6 см.
Тогда МN = √(4² + 6²) = √(16 + 36) = √52 = 2√13 см.

Answer 2

NB1:B1M = NB:BA  - из подобия треугольников NB1M и NBA
выразим NB1
NB1=NB*(B1M:BA)=(NB1+BB1)*(B1M:BA)=(NB1+a)*(1/3)
NB1*(1-1/3)=a*1/3 где а - длина ребра
подставляем а=12
NB1=a*1/2=12/2=6 
по теореме пифагора
NM=корень(6^2+4^2)=корень(52) ~ 7,2 - это ответ
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мое дело решить, оформление оставляю за Вами

Answer 3

To solve this problem, we need to find the point of intersection N between the line AM and the plane formed by the face (BB1CC1) of the cube. We also need to find the length of segment MN, given that the edge of the cube is 12 centimeters.

Let's break down the problem step by step:

Step 1: Find the coordinates of point M

We are given that point M lies on the edge A1B1 of the cube, and MV is equal to 1/3 of A1B1. To find the coordinates of point M, we can use the ratio of lengths.

Since A1B1 is a line segment on the edge of the cube, we can assume that A1 is the origin (0, 0, 0) and B1 has coordinates (12, 0, 0) (since the edge of the cube is 12 centimeters).

Using the ratio MV/A1B1 = 1/3, we can find the coordinates of point M. Let's call the coordinates of point M (x, 0, 0).

The ratio of lengths can be written as: MV/A1B1 = (x - 0)/(12 - 0) = 1/3

Simplifying the equation, we get: x/12 = 1/3

Solving for x, we find: x = 4

Therefore, the coordinates of point M are (4, 0, 0).

Step 2: Find the equation of the line AM

We know that point A has coordinates (0, 0, 0) and point M has coordinates (4, 0, 0). To find the equation of the line AM, we can use the two-point form of a line.

The equation of the line AM can be written as: (x - 0)/(4 - 0) = (y - 0)/(0 - 0) = (z - 0)/(0 - 0)

Simplifying the equation, we get: x/4 = y/0 = z/0

Since y/0 and z/0 are undefined, we can ignore them. Therefore, the equation of the line AM is: x/4 = 1

Step 3: Find the equation of the plane (BB1CC1)

To find the equation of the plane formed by the face (BB1CC1) of the cube, we need to find three non-collinear points on the plane. Let's choose points B, B1, and C.

We know that point B has coordinates (12, 0, 0), point B1 has coordinates (12, 12, 0), and point C has coordinates (12, 12, 12).

Using these three points, we can find the equation of the plane using the point-normal form of a plane.

The equation of the plane (BB1CC1) can be written as: 12(x - 12) + 12(y - 0) + 12(z - 0) = 0

Simplifying the equation, we get: 12x - 144 + 12y + 12z = 0 12x + 12y + 12z = 144

Therefore, the equation of the plane (BB1CC1) is: 12x + 12y + 12z = 144

Step 4: Find the point of intersection N

To find the point of intersection N between the line AM and the plane (BB1CC1), we need to solve the system of equations formed by the line equation and the plane equation.

Substituting the equation of the line AM into the equation of the plane (BB1CC1), we get: 12(4) + 12y + 12z = 144

Simplifying the equation, we get: 48 + 12y + 12z = 144 12y + 12z = 96

Dividing both sides of the equation by 12, we get: y + z = 8

Since we have two variables (y and z) and only one equation, we can choose any value for one variable and solve for the other variable. Let's choose y = 0.

Substituting y = 0 into the equation y + z = 8, we get: 0 + z = 8 z = 8

Therefore, the coordinates of point N are (4, 0, 8).

Step 5: Find the length of segment MN

To find the length of segment MN, we need to calculate the distance between points M and N.

Using the distance formula, the length of segment MN can be calculated as: MN = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Substituting the coordinates of points M (4, 0, 0) and N (4, 0, 8) into the distance formula, we get: MN = sqrt((4 - 4)^2 + (0 - 0)^2 + (8 - 0)^2) MN = sqrt(0 + 0 + 64) MN = sqrt(64) MN = 8

Therefore, the length of segment MN is 8 centimeters.

In conclusion, the point of intersection N between the line AM and the plane (BB1CC1) has coordinates (4, 0, 8), and the length of segment MN is 8 centimeters.