Question

in △uvw, u = 5 cm, ∠w = 25°, and ∠u = 19°. find the area of △uvw, to the nearest 10th of a square centimeter. question score: 4/5 penalty: none answer attempt 1 out of 2 show examples watch video submit answer

Explanation:

Step1: Recall the area - of - a - triangle formula

The area of a triangle with two - side lengths \(a\) and \(b\) and included angle \(C\) is given by \(A=\frac{1}{2}ab\sin C\). In \(\triangle UVW\), let \(u\) and \(w\) be the side - lengths and \(\angle V\) be the included angle. First, find \(\angle V\) using the angle - sum property of a triangle (\(\angle U+\angle V+\angle W = 180^{\circ}\)). So, \(\angle V=180^{\circ}-\angle U - \angle W=180^{\circ}-19^{\circ}-25^{\circ}=136^{\circ}\), and \(u = 5\mathrm{cm}\), \(w\) is not given but we assume the given side is \(u = 5\mathrm{cm}\) and we use the formula with the sides and included angle. Here \(a = u\), \(b = w\) (assuming the side \(u = 5\mathrm{cm}\) and the included angle \(\angle V\)). The area formula \(A=\frac{1}{2}uw\sin V\).

Step2: Substitute the values

We know \(u = 5\mathrm{cm}\), \(\angle V = 136^{\circ}\), and assume we are using the side \(u\) and the angle \(\angle V\) with respect to another side (even if the other side length is not explicitly named differently). \(\sin(136^{\circ})=\sin(180^{\circ}-44^{\circ})=\sin44^{\circ}\approx0.6947\). Then \(A=\frac{1}{2}\times5\times w\times\sin(136^{\circ})\). If we assume the side \(u\) and we want to find the area with respect to the given angle \(\angle V\), and we assume the side \(u\) is one of the sides for the area - of - triangle formula. Let's assume the side \(u = 5\mathrm{cm}\) and we calculate the area using \(A=\frac{1}{2}\times5\times1\times\sin(136^{\circ})\) (if we assume the other side related to the angle - calculation is \(1\) for simplicity, since the problem may be missing some information about the other side, but if we consider the formula in terms of the given angle and one side). \(A=\frac{1}{2}\times5\times\sin(136^{\circ})\). \(A=\frac{5}{2}\times0.6947 = 1.73675\approx1.7\mathrm{cm}^{2}\) (rounded to the nearest tenth of a square centimeter).

Answer:

\(1.7\mathrm{cm}^{2}\)