The sides of the triangle are 4 cm, 15 cm and 13 cm.A perpendicular is drawn through the apex of the smallest angle

The sides of the triangle are 4 cm, 15 cm and 13 cm.A perpendicular is drawn through the apex of the smallest angle to the plane of the triangle, and from its end, which does not belong to the triangle, a 13 cm perpendicular is lowered to the side opposite to this angle. Find the length of the perpendicular to the plane of the triangle.

It is given that KA is the perpendicular to the plane of the triangle ABC. KA =?
In triangle ABC, draw a perpendicular AH.
Consider a triangle ABH. It is rectangular since the AHB angle is 90 degrees. By the Pythagorean theorem:
AB ^ 2 = AH ^ 2 + BH ^ 2
169 = AH ^ 2 + BH ^ 2
BH ^ 2 = 169 – AH ^ 2
Consider AC. AC = 14 cm.
AC = AH + HC
HC = AC – AH
HC = 14 – AH
Consider a triangle AHC, it is rectangular By the Pythagorean theorem:
BC ^ 2 = BH ^ 2 + HC ^ 2
225 = BH ^ 2 + (14-AH) ^ 2
BH ^ 2 = 225 – (14-AH) ^ 2
Let’s make the equation:
169 – AH ^ 2 = 225 – (14-AH) ^ 2
169 – AH ^ 2 = 225 – 196 + 28AH – AH ^ 2
28AH = 140
AH = 5
BH ^ 2 = 169 – AH ^ 2 = 169 – 25 = 144
BH = 12
Consider the triangle KBH, it is rectangular. By the Pythagorean theorem:
KH ^ 2 = BK ^ 2 + BH ^ 2
400 = BK ^ 2 + 144
BK ^ 2 = 256
BK = 16
Answer: 16