In the triangle ABC, AD is the height that divides the base of BC into 2 segments – BD

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

Given: △ ABC, BD = 2 √3 cm, DC = 8 cm, AD – height, ∠ABC = 60 °.
Find: AB, AC.
Since AD ​​is the height △ ABC, the resulting triangles ADC and ADB are rectangular (∠D = 90 °).
If ∠ABC = 60 °, and ∠ADB = 90 °, then, based on the theorem on the sum of the angles of a triangle:
∠BAD = 180 ° – (∠ABC + ∠ADB) = 180 ° – (60 ° + 90 °) = 180 ° – 150 ° = 30 °.
For the property of a right-angled triangle (leg, lying opposite an angle of 30 °):
BD = 1/2 AB.
Hence:
AB = BD * 2 = 2√3 * 2 = 4√3 (cm).
Now, knowing the hypotenuse, we find the AD leg. Behind the Pythagorean theorem:
AB ^ 2 = BD ^ 2 + AD ^ 2.
Hence:
AD ^ 2 = AB ^ 2 – BD ^ 2 = (4 √3) ^ 2 – (2 √3) ^ 2 = (16 * 3) – (4 * 3) = 48 – 12 = 36 (cm).
AD = √36 = 6 (cm).
Now that we know AD, we can find AC. Behind the Pythagorean theorem:
AC ^ 2 = AD ^ 2 + DC ^ 2 = 36 + 64 = 100 (cm).
AC = √100 = 10 (cm).
Answer: AB = 4 √3 cm, AC = 10 cm.