The height BD is drawn in triangle ABC (point D lies on the side AC). Find the perimeter

The height BD is drawn in triangle ABC (point D lies on the side AC). Find the perimeter and area of the triangle if BD = 12, AD = 5 and DC = 16

Given:
triangle ABC,
BD – height,
BD = 12,
AD = 5,
DC = 16.
Find: S ABC and P ABC -?
Solution:
1) Consider the triangle ABC. He has AC = AD + DC;
AC = 5 + 16;
AC = 21;
S ABC = 1/2 * BD * AC;
S ABC = 1/2 * 12 * 21;
S ABC = 6 * 21;
S ABC = 126;
2) Consider the BDC triangle. By the Pythagorean theorem:
BC ^ 2 = BD ^ 2 + DC ^ 2;
ВС ^ 2 = 144 + 256;
BC ^ 2 = 400;
BC = 20;
3) Consider a triangle ВDА. By the Pythagorean theorem:
AB ^ 2 = BD ^ 2 + DA ^ 2;
AB ^ 2 = 144 + 25;
AB ^ 2 = 169;
AB = 13;
4) P ABC = 13 + 20 + 21 = 54.
Answer: S ABC = 126 and P ABC = 54.