Question

solve the right triangle.
write your answers as integers or as decimals rounded to the nearest tenth.
id =
hi =
m∠j = °

Explanation:

Step1: Find \(m\angle J\)

In a right - triangle, the sum of the two non - right angles is \(90^{\circ}\). Given \(\angle H = 42^{\circ}\), then \(m\angle J=90^{\circ}-\angle H\).
\(m\angle J = 90 - 42=48^{\circ}\)

Step2: Find \(HI\)

Using the sine function: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle J = 48^{\circ}\), \(\sin J=\frac{HI}{JH}\). Given \(JH = \sqrt{82}\approx9.06\).
\(\sin48^{\circ}=\frac{HI}{\sqrt{82}}\), so \(HI=\sqrt{82}\sin48^{\circ}\).
Since \(\sin48^{\circ}\approx0.743\), \(HI\approx9.06\times0.743\approx6.7\)

Step3: Find \(IJ\)

Using the cosine function: \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle J = 48^{\circ}\), \(\cos J=\frac{IJ}{JH}\).
\(\cos48^{\circ}=\frac{IJ}{\sqrt{82}}\), so \(IJ=\sqrt{82}\cos48^{\circ}\).
Since \(\cos48^{\circ}\approx0.669\), \(IJ\approx9.06\times0.669\approx6.1\)

Answer:

\(m\angle J = 48^{\circ}\), \(HI\approx6.7\), \(IJ\approx6.1\)