In a triangle ABC AC = BC = 4√5, AB = 16. Find tgA.

Let’s build the height of the CH to the AB side.

Since, by condition, AC = BC, the triangle ABC is isosceles, and then its height CH is also the median and bisector.

Then AH = BH = AB / 2 = 16/2 = 8 cm, and the triangles ACН and BCH are rectangular.

In a right-angled triangle AСН, according to the Pythagorean theorem, we determine the length of the leg CH.

CH ^ 2 = AC ^ 2 – AH ^ 2 = 80 – 64 = 16.

CH = 4 cm.

In a right-angled triangle, the tangent of an acute angle is the ratio of the opposite leg to the adjacent one.

tgA = CH / AC = 4/4 * √5 = 1 / √5 = √5 / 5.

Answer: The tangent of angle A is √5 / 5.