Question

triangle xyz has vertices x(-1,-1), y(-2,1), and z(1,2). what is the approximate measure of angle z?
37.2°
61.5°
78.5°
81.3°

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

• For \(XY\): \(x_1=-1,y_1 = - 1,x_2=-2,y_2 = 1\)

\(XY=\sqrt{(-2 + 1)^2+(1 + 1)^2}=\sqrt{1 + 4}=\sqrt{5}\)

• For \(YZ\): \(x_1=-2,y_1 = 1,x_2=1,y_2 = 2\)

\(YZ=\sqrt{(1 + 2)^2+(2 - 1)^2}=\sqrt{9+1}=\sqrt{10}\)

• For \(XZ\): \(x_1=-1,y_1 = - 1,x_2=1,y_2 = 2\)

\(XZ=\sqrt{(1 + 1)^2+(2 + 1)^2}=\sqrt{4 + 9}=\sqrt{13}\)

Step2: Use the cosine law

The cosine law is \(\cos Z=\frac{YZ^{2}+XZ^{2}-XY^{2}}{2\cdot YZ\cdot XZ}\)
Substitute \(XY = \sqrt{5}\), \(YZ=\sqrt{10}\), \(XZ=\sqrt{13}\)
\(\cos Z=\frac{10 + 13-5}{2\cdot\sqrt{10}\cdot\sqrt{13}}=\frac{18}{2\sqrt{130}}=\frac{9}{\sqrt{130}}\approx0.786\)

Step3: Find the angle

\(Z=\cos^{-1}(0.786)\approx37.2^{\circ}\)

Answer:

\(37.2^{\circ}\)