Question

Медианы треугольника МNK пересекаются в точке О. Через точку О проведена прямая, параллельная стороне МК и пересекающая стороны МN и NК в точках А и В соответственно. Найдите МК, Если длина отрезка АВ равна 12 см.

Answer 1

пересекающая стороны МN (не MK наверно) и NK в точках А и В соответственно 
AB/MK = 2/(2+1) (треугольники ANB и MNK подобны; медианы точкой пересечения (точка O) делятся в отношении 2 к 1, считая от вершины) 
MK = (3·AB)/2 = 18 (см)

Answer 2

Problem Analysis

We are given a triangle MNK, where the medians intersect at point O. A line is drawn through point O, parallel to side MK, and it intersects sides MN and NK at points A and B, respectively. The length of segment AB is given as 12 cm. We need to find the length of segment MK.

Solution

To find the length of segment MK, we can use the property of medians in a triangle. The medians of a triangle divide each other in a 2:1 ratio. This means that the length of segment MO is twice the length of segment AO, and the length of segment KO is twice the length of segment BO.

Let's assume the length of segment AO as x. Then, the length of segment BO is also x, as they are divided in a 2:1 ratio. Since the length of segment AB is given as 12 cm, we can write the equation:

x + x = 12

Simplifying the equation, we get:

2x = 12

Dividing both sides by 2, we find:

x = 6

Therefore, the length of segment AO and segment BO is 6 cm each. Since the length of segment MO is twice the length of segment AO, and the length of segment KO is twice the length of segment BO, we can conclude that:

MO = 2 * AO = 2 * 6 = 12 cm KO = 2 * BO = 2 * 6 = 12 cm

Since point O is the intersection of the medians, it is also the centroid of the triangle. The centroid divides each median in a 2:1 ratio. Therefore, the length of segment MK is three times the length of segment KO:

MK = 3 * KO = 3 * 12 = 36 cm

Hence, the length of segment MK is 36 cm.

Answer

The length of segment MK is 36 cm.