The height of an isosceles triangle drawn to the side divides it into 4 cm and 16 cm segments

The height of an isosceles triangle drawn to the side divides it into 4 cm and 16 cm segments, counting from the apex of the apex angle. Find the perimeter of the triangle.

Triangle ABC, AB = BC, AH – height, BH = 4 cm, CH = 16 cm.
1. Let’s find the length of the lateral side of the triangle ABC:
BC = BH + CH;
BC = 4 + 16 = 20 (cm).
Then:
AB = BC = 20 cm.
2. Consider the triangle AHB: angle AHB = 90 degrees, AB = 20 cm – hypotenuse, BH = 4 cm and AH – legs.
By the Pythagorean theorem:
AH = √ (AB ^ 2 – BH ^ 2) = √ (20 ^ 2 – 4 ^ 2) = √ (400 – 16) = √384 = 8√6 (cm).
3. Consider a triangle AHC: angle AHC = 90 degrees, AU – hypotenuse, AH = 8√6 cm and CH = 16 cm – legs.
By the Pythagorean theorem:
AC = √ (AH ^ 2 + CH ^ 2) = √ ((8√6) ^ 2 + 16 ^ 2) = √ (384 + 256) = √640 = 8√10 (cm).
4. Perimeter is the sum of the lengths of all sides:
P = AB + BC + AC;
P = 20 + 20 + 8√10 = 40 + 8√10 (cm).
Answer: P = 40 + 8√10 cm.