Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. a = 10.8, b = 24.7

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Identify the sides and angles in the right triangle. Here, side \(p\) is opposite angle \(P\), side \(q\) is adjacent to angle \(P\), and side \(r\) is the hypotenuse opposite the right angle at \(R\).

Use the Pythagorean theorem to find the length of the hypotenuse \(r\). The formula is \(r = \sqrt{p^2 + q^2}\), where \(p = 10.8\) and \(q = 24.7\).

Calculate angle \(P\) using the tangent function, since you know the opposite side \(p\) and adjacent side \(q\). Use \(\tan(P) = \frac{p}{q}\), then find \(P = \arctan\left(\frac{p}{q}\right)\).

Calculate angle \(Q\) by subtracting angle \(P\) from 90 degrees, because the sum of angles in a right triangle is 90 degrees for the two non-right angles: \(Q = 90^\circ - P\).

Round the lengths and angles to the required precision: lengths to two decimal places and angles to the nearest tenth of a degree.

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