# Through point A, inclined AB, AC and perpendicular OA.AB = 2a are drawn to plane a. The angles between straight

Through point A, inclined AB, AC and perpendicular OA.AB = 2a are drawn to plane a. The angles between straight lines AB, AC and plane a are respectively 30 degrees and 45 degrees. Find the lengths of the perpendicular AO, inclined AC and its projection

As we know from the condition: the perpendicular AO, oblique AB and AC and their projections OB and OS form right-angled triangles AOB and BOС (angle O is right).
Since the angle OCA = 45⁰, the triangle AOC is a rectangular isosceles, therefore, the perpendicular AO is equal to its projection AO.
In triangle AOB, angle ABO = 30⁰.
In a right-angled triangle, the leg, which lies opposite an angle of 30⁰, is 2 times less than the hypotenuse, we have AO = 2a / 2 = a, and AO = OC = a.
From the triangle AOC it follows: AC = a / sin45⁰ = a√2.
Answer: AO = a, OC = a, AC = a√2.

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