Question

find the smallest angle of $\triangle wxy$. triangle with sides: $wx = 37$ m, $wy = 38$ m, $xy = 31$ m

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the smallest angle is opposite the shortest side. First, we identify the lengths of the sides of \(\triangle WXY\): \(WX = 37\space m\), \(XY=31\space m\), \(WY = 38\space m\). The shortest side is \(XY = 31\space m\), and the angle opposite to \(XY\) is \(\angle W\).

Step2: Use the Law of Cosines to find \(\angle W\)

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite to side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = WX=37\), \(b = WY = 38\), \(c=XY = 31\). We want to find \(\angle W\) (opposite to \(XY\)), so we can rewrite the Law of Cosines as:
\(\cos W=\frac{WX^{2}+WY^{2}-XY^{2}}{2\cdot WX\cdot WY}\)
Substitute the values: \(WX = 37\), \(WY=38\), \(XY = 31\)
\(WX^{2}=37^{2}=1369\), \(WY^{2}=38^{2} = 1444\), \(XY^{2}=31^{2}=961\)
\(\cos W=\frac{1369 + 1444-961}{2\times37\times38}=\frac{1369+1444 - 961}{2812}=\frac{2813 - 961}{2812}=\frac{1852}{2812}\approx0.6586\)

Step3: Find the angle \(W\)

To find the angle whose cosine is approximately \(0.6586\), we use the inverse cosine function: \(\angle W=\cos^{- 1}(0.6586)\approx49.8^{\circ}\) (We can also check the other angles for verification. The side \(WX = 37\) is opposite \(\angle Y\), and \(WY = 38\) is opposite \(\angle X\). Since \(XY\) is the shortest side, \(\angle W\) should be the smallest angle.)

Answer:

The smallest angle of \(\triangle WXY\) is approximately \(50^{\circ}\) (or more precisely, approximately \(49.8^{\circ}\))