In a right-angled triangle with a 30-degree angle, the opposite side (PQ) is 12, and the hypotenuse (PR) is 24.
In the given right-angled triangle, with a 90-degree angle at Q and an angle R measuring 30 degrees, the base RQ is known to be 12√3. The task is to determine the unknown side lengths, PQ (the opposite side) and PR (the hypotenuse).
Applying trigonometry in a right-angled triangle involves using the tangent function, as it relates the opposite and adjacent sides. For angle R (30 degrees), the equation is tan(R) = opposite side / adjacent side. Solving for PQ, we get PQ = RQ * tan(30 degrees).
The tangent of 30 degrees is 1/√3. Substituting this value, we find PQ = 12√3 * (1/√3) = 12.
Moving on to find PR, the hypotenuse, we apply the Pythagorean theorem: PR^2 = PQ^2 + RQ^2. Substituting the values, we get PR^2 = 12^2 + (12√3)^2 = 576, leading to PR = √576 = 24.
In summary, PQ is 12, and PR is 24 in the right-angled triangle with a 30-degree angle.