Точка M середина стороны AC треугольника ABC в котором AB=6см Периметры треугольников ABM и BCM отличаются на 10 см найти сторону BC

Let's denote the lengths of the sides of triangle ABC as follows: AB = 6 cm, BC = x cm, and AC = y cm.

We know that point M is the midpoint of side AC. Therefore, AM = MC = y/2 cm.

The perimeter of triangle ABC is the sum of the lengths of its sides, so we can write the following equation:

AB + BC + AC = 6 + x + y = 2(AM + MC) = 2(y/2 + y/2) = 2y.

Simplifying this equation, we have: 6 + x + y = 2y, 6 + x = y.

We are also given that the perimeters of triangles ABM and BCM differ by 10 cm. The perimeter of triangle ABM is AB + BM + AM, and the perimeter of triangle BCM is BC + BM + MC. Therefore, we can write the following equation:

(AB + BM + AM) - (BC + BM + MC) = 10.

Substituting the known values, we have: (6 + BM + (y/2)) - (x + BM + (y/2)) = 10, 6 + y/2 - x = 10, y/2 - x = 4, x = y/2 - 4.

Now we can substitute this value of x into the equation 6 + x = y to find the value of y:

6 + (y/2 - 4) = y, 12 + y - 8 = 2y, 4 = y.

So, the length of side BC is 4 cm.