Medians AD and BE are drawn in triangle ABC. Find: perimeter of triangle ABC if AB = 8cm, CD = 2cm, AE = 4cm.

Take triangle ABC with sides AB; AC and BC. Let us denote by a; b and from the length of the sides of this triangle:

a = | BC |;

b = | AC |;

c = | AB |;

Let’s designate the middle of side BC as point D, and the middle of side AC – as point E. Connecting point D with the apex of triangle A, we get median AD to side BC, and connecting point E with vertex B, we get median BE to side of triangle AC. It is known that:

c = | AB | = 8 cm;

a1 = | CD | = 2 cm;

b1 = | AE | = 4 cm;

The task requires to find the perimeter P of the triangle ABC.

Equation for the perimeter of triangle ABC
To solve this problem:

we write the equality for the perimeter P of the triangle;
calculate the lengths of the sides a and b of the triangle ABC;
substitute the obtained values ​​for the lengths of the sides and calculate the perimeter P.
The perimeter of a triangle is the sum of the lengths of all its sides:

P = a + b + c;

Point D bisects side BC. It means that:

| BD | = | CD |

and correspondingly:

a = | BC | = | BD | + | CD | = 2 * | CD | = 2 * a1;

Taking into account that the point E divides the side AC in half, we similarly obtain that:

b = | AC | = | AE | + | CE | = 2 * | AE | = 2 * b1;

Substituting these expressions into the formula for the perimeter of a triangle, we have:

P = a + b + c = 2 * a1 + 2 * b1 + c;

Calculating the perimeter of a triangle ABC
Let us further substitute the initial values ​​according to the text of the problem into the obtained equality. We calculate the perimeter:

P = 2 * a1 + 2 * b1 + c = 2 * 2 + 2 * 4 + 8 = 4 + 8 + 8;

P = 20 (cm);

Answer: the perimeter of triangle ABC is 20 cm.