The perimeter of triangle ABC is 36 cm AB = BC BD median. Find BD if the perimeter of the CBD triangle is 30cm.

Consider triangles ABD and CBD: AB = BC (by condition), AD = BD (BD median), BD is common, triangles are equal, which means that their perimeters are also equal.

Let’s express the perimeter of each triangle:

P (ABD) = AB + AD + BD = 30 (cm),

P (CBD) = BC + BD + CD = 30 (cm).

Let’s add these perimeters:

P (ABD) + P (CBD) = AB + AD + BD + BC + BD + CD = AB + BC + (AD + CD) + 2BD.

Since AD + CD = AC, it turns out: P (ABD) + P (CBD) = (AB + BC + AC) + 2BD.

But (AB + BC + AC) is the perimeter of triangle ABC, it turns out:

P (ABD) + P (CBD) = P (ABC) + 2BD.

We substitute known data.

30 + 30 = 36 + 2BD.

2BD = 60 – 36;

2BD = 24;

BD = 24: 2 = 12 (cm).

Answer: BD = 12 cm.