In an isosceles triangle ABC, it is known that BC = CA = 8√3. What is the height dropped to AB if tg A = √3?

1. Let us denote by HH the height drawn to the side AB of the triangle ABC.

2. The angle BAC, the tangent of which is √3, is 60 °.

3. According to the properties of an isosceles triangle, the angles at its base are equal, that is

angle BAC = ABC = 60 °.

4. Angle АСВ = 180 ° – 60 ° – 60 ° = 60 °.

5. All angles of the triangle are equal, therefore it is equilateral:

AB = BC = AC = 8√3 cm.

6. We calculate the length of the CH height through the sine of the angle BAC:

CH / AC = sine 60 = √3 / 2.

CH = AC x √3 / 2 = 8√3 x √3 / 2 = 12 cm.

Answer: height CH = 12 cm.