Find the measure of $\angle J$, the smallest angle in a triangle with sides measuring 11, 13, and 19. Round to the nearest whole degree.
To find the measure of the smallest angle \(\angle J\) in a triangle with side lengths \(a=11\), \(b=13\), and \(c=19\), we first identify the side opposite to the smallest angle. In any triangle, the smallest angle is always opposite the shortest side. Here, the side lengths are \(11\), \(13\), and \(19\). The shortest side is \(a=11\), so the smallest angle \(\angle J\) is the angle opposite side \(a\).
We use the Law of Cosines, which relates the sides of a triangle to the cosine of one of its angles:
\[a^2 = b^2 + c^2 - 2bc \cos(J)\]
Rearranging this formula to solve for \(\cos(J)\), we get:
\[\cos(J) = \frac{b^2 + c^2 - a^2}{2bc}\]
Substituting the known values \(a=11\), \(b=13\), and \(c=19\):
\[\cos(J) = \frac{13^2 + 19^2 - 11^2}{2 \times 13 \times 19}\]
\[\cos(J) = \frac{169 + 361 - 121}{494}\]
\[\cos(J) = \frac{409}{494}\]
\[\cos(J) \approx 0.8279\]
Now, we find the measure of \(\angle J\) by taking the inverse cosine (arccos) of the calculated value:
\[J = \arccos(0.8279)\]
Using the calculation, we find \(J \approx 34.11^\circ\).
Rounding this value to the nearest whole degree gives \(34^\circ\). Thus, the smallest angle \(\angle J\) is \(34^\circ\).
When designing structural supports for a triangular roof, architects often need to calculate the precise angles at which beams meet to ensure stability. If a support beam connects two points such that the side lengths of the formed triangle are 11m, 13m, and 19m, finding the smallest angle (34°) helps the carpenter cut the wood at the exact required joint angle for a perfect fit. This ensures that the structure remains balanced and durable under different weather conditions.