Question

Срочно!!!Медиана СМ прямоугольного треугольника АВС равна 12 и делит прямой угол в отношении ACM : BCM = 1:2. Найти катет ВС.

Answer 1

Медиана проведённая из прямого угла к гипотенузе делит гипотенузу пополам и равна половине гипотенузы из этого следует что СМ=МВ=12
угол АСМ=90/(1+2)*1=90/3*1=30*1=30
угол ВСМ=90-30=60
вспоминаем что СМ=МВ=12 значит треугольник СМВ-равнобедренный значит угол ВСМ=углу СВМ=60  ну а если в треугольнике два угла по 60 градусов значит и третий равен 60 градусов значит треугольник СМВ ещё и равносторонний а значит что ВС=СМ=МВ=12

Answer 2

Problem Statement

You are given a right-angled triangle ABC, where the median CM is equal to 12 and divides the right angle in the ratio ACM : BCM = 1:2. You need to find the length of the side BC.

Solution

To find the length of side BC, we can use the properties of medians in a triangle.

Let's denote the length of side AC as a, the length of side BC as b, and the length of side AB as c.

Since CM is the median, it divides the side AB into two equal parts. Let's denote the length of AM as x. Therefore, the length of MB is also x.

According to the given information, the ratio of ACM to BCM is 1:2. This means that the length of AM is one-third of CM, and the length of MB is two-thirds of CM.

We are also given that the median CM is equal to 12. Therefore, the length of AM is 12/3 = 4, and the length of MB is 12/3 * 2 = 8.

Now, let's consider the right-angled triangle AMC. We can use the Pythagorean theorem to find the length of side AC:

AC^2 = AM^2 + CM^2

Substituting the values, we get:

a^2 = 4^2 + 12^2

Simplifying, we have:

a^2 = 16 + 144

a^2 = 160

Taking the square root of both sides, we get:

a = √160

Simplifying further, we have:

a = 4√10

Now, let's consider the right-angled triangle BMC. Again, we can use the Pythagorean theorem to find the length of side BC:

BC^2 = BM^2 + CM^2

Substituting the values, we get:

b^2 = 8^2 + 12^2

Simplifying, we have:

b^2 = 64 + 144

b^2 = 208

Taking the square root of both sides, we get:

b = √208

Simplifying further, we have:

b = 4√13

Therefore, the length of side BC is 4√13.

Answer

The length of side BC in the right-angled triangle ABC is 4√13.