In triangle ABC, height AD divides the base of BC into segments BD = 2√3 cm and

In triangle ABC, height AD divides the base of BC into segments BD = 2√3 cm and DC = 8 cm, angle ABC = 60 degrees. Find the sides of the triangle.

Consider a right-angled triangle ADB (AD – conditional height).
∠ DAB = 90 ° – ∠ ABC = 90 ° – 60 ° = 30 °.
The BD leg is located opposite an angle of 30 °, which means the hypotenuse AB = 2 * BD = 4√3 (cm).
In the same triangle, according to the Pythagorean theorem, we find leg AD:
AD = √ (AB² – BD²) = √ (48 – 12) = √36 = 6 (cm).
Consider a right-angled triangle ADC, two legs are known in it, we find the hypotenuse AC:
AC = √ (AD² + DC²) = √ (36 + 64) = √100 = 10 (cm).
Answer: the sides of the triangle are 4√3 cm and 10 cm.