Question

Катеты СВ и СА прямоугольного треугольника АВС равны 9 и 12 соответственно. М - точка пересечения медиан. Найдите длину отрезка СМ.

Answer 1

По Пифагору находим гипотенузу АВ.
АВ = √(81+144) = √225 = 15.
Медиана СД равна половине гипотенузы: СД = 15/2 = 7,5.
Точка М делит СД в отношении 2:1 от точки С.
Приходим к ответу: СМ = 7,5*(2/3) = 5.

Answer 2

Task: Finding the length of segment CM in a right triangle ABC with given catheti

To find the length of segment CM in a right triangle ABC with given catheti, we can use the properties of medians in a triangle.

Let's start by understanding the properties of medians in a triangle:

- A median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. - The medians of a triangle intersect at a point called the centroid, which divides each median into two segments in a 2:1 ratio. - The centroid divides each median into two segments: one segment from the vertex to the centroid, and another segment from the centroid to the midpoint of the opposite side.

In our case, M is the point of intersection of the medians, and we need to find the length of segment CM.

Solution:

To find the length of segment CM, we can use the property that the centroid divides each median into two segments in a 2:1 ratio.

Let's denote the length of segment CM as x. According to the 2:1 ratio, we can write:

CM = 2 * XM (1)

To find the length of XM, we can use the property that the centroid divides each median into two segments in a 2:1 ratio. Let's denote the length of segment AM as a. According to the 2:1 ratio, we can write:

AM = 2 * XM (2)

Now, let's find the length of segment AM. We know that AM is a median, which means it divides the side BC into two equal segments. Let's denote the length of segment BC as b. According to this information, we can write:

BC = 2 * AM (3)

We are given the lengths of the catheti AB and AC as 9 and 12, respectively. Since AB and AC are the sides of a right triangle, we can use the Pythagorean theorem to find the length of the hypotenuse BC:

BC = sqrt(AB^2 + AC^2) (4)

Substituting the given values, we have:

BC = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 (5)

Now, we can substitute the value of BC into equation (3) to find the length of segment AM:

15 = 2 * AM (6)

Simplifying equation (6), we have:

AM = 15 / 2 = 7.5 (7)

Finally, we can substitute the value of AM into equation (2) to find the length of segment XM:

7.5 = 2 * XM (8)

Simplifying equation (8), we have:

XM = 7.5 / 2 = 3.75 (9)

Now, we can substitute the value of XM into equation (1) to find the length of segment CM:

CM = 2 * XM = 2 * 3.75 = 7.5

Therefore, the length of segment CM is 7.5.

Answer:

The length of segment CM in the right triangle ABC with catheti AB = 9 and AC = 12 is 7.5.