Question

Плоскость а перпендикулярна к катету МК прямоугольного треугольника MNK и делит его в отношении ММ1:М1К=3:2. В каком отношении плоскость а делит гипотенузу МN

Answer 1

Плоскость параллельна катету KN. Треугольники подобны-> 
-> в таком же отношении.

Answer 2

Плоскость а делит гипотенузу MN в отношении MM1:M1K = 3:2.

Understanding the problem

To understand how the plane divides the hypotenuse MN, let's first analyze the given information. We have a right-angled triangle MNK, where the plane a is perpendicular to the cathetus MK. The plane divides the triangle in the ratio MM1:M1K = 3:2.

Solution

To find the ratio in which the plane divides the hypotenuse MN, we can use the concept of similar triangles. Since the plane is perpendicular to the cathetus MK, it will also be perpendicular to the hypotenuse MN. This means that the triangles MM1N and M1KN are similar.

Let's denote the length of the hypotenuse MN as x. Since the ratio MM1:M1K is given as 3:2, we can express the lengths of MM1 and M1K in terms of x. Let MM1 be 3a and M1K be 2a, where a is a constant.

According to the properties of similar triangles, the corresponding sides of similar triangles are proportional. Therefore, we can set up the following equation:

(MM1 / MN) = (M1K / KN)

Substituting the values, we get:

(3a / x) = (2a / (x - 2a))

Now, we can solve this equation to find the value of x, which represents the length of the hypotenuse MN.

Cross-multiplying the equation, we get:

3a * (x - 2a) = 2a * x

Expanding and simplifying the equation, we get:

3ax - 6a^2 = 2ax

Rearranging the terms, we get:

ax = 6a^2

Dividing both sides by a, we get:

x = 6a

Therefore, the length of the hypotenuse MN is 6 times the constant a.

Now, let's find the ratio in which the plane a divides the hypotenuse MN. Since the length of MM1 is 3a and the length of M1K is 2a, the plane divides the hypotenuse MN in the ratio MM1:M1K = 3:2.

In conclusion, the plane a divides the hypotenuse MN in the ratio 3:2.

I hope this explanation helps! Let me know if you have any further questions.