In the parallelogram КРВА (angle К acute), the diagonal PA and the height of the PH are drawn from the vertex Р.

In the parallelogram КРВА (angle К acute), the diagonal PA and the height of the PH are drawn from the vertex Р. Find RA and AN, if PH = 9, the RAS angle is 60 degrees.

Since PH – height, the PHA triangle is rectangular. In a right-angled triangle, the sum of the acute angles is 90, that is, the RAS angle + the NRA angle = 90. The angle of RAS = 60, which means the angle of NRA = 30. By the property of a right-angled triangle: opposite an angle of 30 degrees lies a leg equal to half of the hypotenuse. So HA = RA / 2 or RA = 2 * HA. Let’s apply the Pythagorean theorem:
PA ^ 2 = PH ^ 2 + HA ^ 2
(2 * HA) ^ 2 = 9 ^ 2 + HA ^ 2
4 * HA ^ 2 = 81 + HA ^ 2
4 * HA ^ 2-HA ^ 2 = 81
3 * HA ^ 2 = 81
HA ^ 2 = 27
HA = √ (27)
HA = 3 * √ (3)
Then RA = 2 * HA = 2 * 3 * √ (3) = 6 * √ (3)
Answer: HA = 3 * √ (3), RA = 6 * √ (3)