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.