In triangle ABC, side AB is 3 cm, BC is 5 cm and AC = 4 cm. Find the smallest height of the triangle.

1. Calculate the area (S) of a triangle using the formula of Heron’s theorem:

S = √p (p – AB) (p – BC) (p – AC).

p – semi-perimeter = (3 + 4 + 5) / 2 = 6 cm.

S = √6 (6 – 3) (6 – 5) (6 – 4) = √6 x 3 x 1 x 2 = √36 = 6 cm².

2. Let’s draw the heights of ВK, AР, CE to the sides of AC, BC, AB, respectively.

3. We calculate their lengths using another formula for the area of a triangle:

S = AC x ВK / 2. ВK = 2 x 6/4 = 3 cm.

S = BC x AP / 2. AР = 2 x 6/5 = 2.4 cm.

S = AB x CE / 2. CE = 2 x 6/3 = 4 cm.

Answer: AР = 2.4 cm – the smallest height of the triangle.