The median BD is drawn in an isosceles triangle ABC with base AC. The perimeter of triangle ABD is 36 cm

The median BD is drawn in an isosceles triangle ABC with base AC. The perimeter of triangle ABD is 36 cm, and the perimeter of triangle ABC is 64 cm. Find the length of BD.

P ABC = 64 cm, P ABD = 36 cm.

The perimeter of a triangle is the sum of the lengths of all its sides. Thus, we represent the perimeter of triangle ABC through its sides:

P ABC = AB + BC + AC;

Considering that ABC is an isosceles triangle, i.e. AB = BC, then P ABC = 2 * AB + AC. Since BD is the median to the base AC, it means that it divides it into two equal parts, i.e. AD = DC. Thus, we can write:

P ABC = 2 * AB + 2 * AD;

2 * AB + 2 * AD = 64;

AB + AD = 64: 2;

AB + AD = 32.

Let us express the perimeter of triangle ABD:

P ABD = AB + BD + AD;

Substitute the resulting sum AB + AD = 32, and we get: P ABD = (AB + AD) + BD = 32 + BD;

That is, 32 + BD = 36;

BD = 36 -32 = 4 cm.

Answer: The median length BD of triangle ABC is 4 cm.