The bisector BK is drawn in the triangle ABC. Determine the length of the BC

The bisector BK is drawn in the triangle ABC. Determine the length of the BC side if it is known that AK = 5, CK = 3, and the perimeter of the ABC triangle is 20.

Knowing the length of the segments into which the side ac is divided by the bisector, we find ac:

ac = ak + ck = 5 + 3 = 8 cm;

So, one side of the triangle is 8 cm.

Therefore, the sum of the sides ab and bc is equal to:

20 – 8 = 12 cm;

Let’s make a proportion
One of the properties of the bisector is that the line segments into which the opposite side is divided are directly proportional to the other two sides of the triangle. That is:

ak / ab = ck / bc;

5 / ab = 3 / bc;

Since ab + bc = 12, then ab = 12 – bc;

Let’s write the proportion by replacing ab:

5 / (12 – bc) = 3 / bc;

Equation solution
So, we got an equation with one unknown bc.

Let’s solve it:

5 / (12 – bc) = 3 / bc;

Let’s get rid of the denominators:
5 * bc = 3 * (12 – bc);

Let’s expand the brackets:
5 * bc = 36 – 3 * bc;

We move the unknowns to the left side of the equation:
5 * bc + 3 * bc = 36;

Let’s simplify:
8 * bc = 36;

Let’s find the unknown:
bc = 36/8 = 4.

So, side bc is 4 cm.

Answer: 4 cm.