Find the acute angles of a right triangle if its hypotenuse is 16 and its area is 32√3

Draw the median and height from the right angle, denoting them m and h, respectively.
If we describe a circle around a triangle, then the center of this circle will lie in the middle of the hypotenuse (by the theorem on the circumscribed circle). Hence:
m = c / 2 = 16/2 = 8
S = (1/2) hc => h = 2S / c = 2 * 32√3 / 16 = 4√3
By definition of the sine:
sinβ = h / m = 4√3 / 8 = √3 / 2
According to the table, we determine that β = 60 °
The angle γ is external to β, therefore γ = 180 ° -β = 180 ° -60 ° = 120 °
A triangle containing an angle γ isosceles, since the median m and half of the hypotenuse are equal (we found out earlier).
Therefore, by the property of an isosceles triangle, the angles at the base are equal (denoted by α).
Then, by the theorem on the sum of the angles of a triangle:
180 ° = γ + α + α
180 ° = 120 ° + 2α
α = 30 ° is one of the desired angles.
We find another desired angle by the same theorem on the angles of a triangle: 180 ° -90 ° -30 ° = 60 °
answer: 30 ° and 60 °

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