# In a regular quadrangular pyramid PABCD with base ABCD AB = 2√2, PC = √39. K is the middle

In a regular quadrangular pyramid PABCD with base ABCD AB = 2√2, PC = √39. K is the middle of AB, N is the middle of AD. Find the tangent of the PNK angle.

Let us determine the lengths of the diagonals BD and AC.

BD ^ 2 = AC ^ 2 = 2 * AB ^ 2 = 2 * (2 * √2) ^ 2 = 16.

BD = AC = 4 cm.

Since points K and N are the middle of the edges AB and AD, then the segment KN is the middle line of the triangle ABD, then KN = BD / 2 = 4/2 = 2 cm.Then HK = HN = KN / 2 = 2/2 = 1 cm …

The diagonal AC is divided in half at the point O, then AO = AC / 2 = 4/2 = 2 cm, and OH = AO / 2 = 2/2 = 1 cm.

The ABP triangle is isosceles, then the PK is its height and median, which means that AK = BK = AB / 2 = 2 * √2 / 2 = √2 cm.

Then, by the Pythagorean theorem, RK ^ 2 = AP ^ 2 – AK ^ 2 = 39 – 2 = 37. PK = √37 cm.

In a right-angled triangle PNA, according to the Pythagorean theorem, PH ^ 2 = PK ^ 2 – HK ^ 2 = 37 – 1 = 36.

HP = 6 cm.

Determine the tangent of the angle PNK.

tgPNK = PH / HN = 6/1 = 6.

Answer: The tangent of the PNK angle is 6. One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.