Let’s denote by x the length of the leg of this right-angled triangle, which lies opposite the angle of 30 °, and by y – the length of the second leg.
Using the formulas of a right-angled triangle, we can express the length of the second leg through the length of the first leg:
y = x * tan 30 ° = x / √3.
By the condition of the problem, the area of this right-angled triangle is 32√3.
Since the area of any right-angled triangle is half the product of its legs, we can write the following equation:
(1/2) * x * x / √3 = 32√3.
Solving this equation, we get:
x ^ 2 / (2√3) = 32√3;
x ^ 2 = 32√3 * 2√3;
x ^ 2 = 32 * 2 * 3;
x ^ 2 = 64 * 3;
x = √ (64 * 3);
x = 8√3.
Answer: the length of the leg opposite the 30 ° angle is 8√3.
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