In a triangle ABC AC = BC = 2√3, angle C is 120 degrees. Find the height of AH.

In the ABC triangle it is known:

AC = BC = 2√3;
Angle C = 120 °;
Find the height of AН.

Decision:

1) If 2 sides in a triangle are equal, then the triangle is isosceles.

2) The height AH is perpendicular to the BC side.

3) Angle A = angle B;

Angle C = 120 °;

Then angle A + angle B = 180 ° – 120 ° = 60 °;

Angle B = 60 ° / 2 = 30 °;

4) Consider triangle АНВ with right angle H.

HAB angle = 180 ° – 90 ° – 30 ° = 90 ° – 30 ° = 60 °;

5) The angle opposite 30 ° is equal to half of the hypotenuse.

6) СK – the height of the triangle ABC.

cos 60 = AK / AC;

AK = 2√3 * 1/2 = √3;

7) AB = 2√3;

8) ВН = 2√3 / 2 = √3;

9) AH = √ (AB ^ 2 – BH ^ 2) = √ ((2√3) ^ 2 – √3 ^ 2) = √ (4 * 3 – 9) = √ (12 – 9) = √3.