The perimeter of a triangle is 15 cm, the median of this triangle divides it into 2 triangles

The perimeter of a triangle is 15 cm, the median of this triangle divides it into 2 triangles with the following perimeters: 11 cm and 14 cm. Find the length of the median

Let us denote the sides of the triangle by the letters a, b and c, and the median by the letter x (suppose that it is drawn to side c and divides it into parts c / 2 and c / 2).

Let us express the perimeter of the triangle: a + b + c = 15 (cm).

Let us express the perimeters of the parts of the triangle: a + x + c / 2 = 14 and b + x + c / 2 = 11 (cm).

We represent the last two equations as a system and solve it by the addition method (subtract the second from the first equation):

(a + x + c / 2) – (b + x + c / 2) = 14 – 11;

a + x + c / 2 – b – x – c / 2 = 3;

a – b = 3.

Let us express the variable a from here: a = 3 + b.

Substitute the expressed value of a into the equation for the perimeter of the triangle a + b + c = 15:

(3 + b) + b + c = 15;

3 + b + b + c = 15;

3 + 2b + c = 15;

2b + c = 15 – 3;

2b + c = 12;

s = 12 – 2v;

hence c / 2 = (12 – 2v) / 2 = 6 – c.

Substitute the expressed values ​​of a and c / 2 into the equation a + x + c / 2 = 14:

(3 + b) + x + (6 – b) = 14;

3 + b + x + 6 – b = 14;

9 + x = 14;

x = 14 – 9;

x = 5.

Answer: The median of the triangle is 5 cm.