Question

solve the right triangle
write your answers as integers or as decimals rounded to the nearest tenth.
de =
cd =
m∠e = °

Explanation:

Step1: Find \(m\angle E\)

In a right - triangle, the sum of the two non - right angles is \(90^{\circ}\).
If \(\angle C = 31^{\circ}\) and \(\angle D=90^{\circ}\), then \(m\angle E=90^{\circ}-m\angle C\).
\(m\angle E = 90 - 31=59^{\circ}\)

Step2: Find \(CD\)

We know that \(\sin E=\frac{CD}{CE}\). First, \(CE=\sqrt{78}\approx8.83\).
Since \(\sin E=\sin(59^{\circ})\approx0.857\) and \(\sin E=\frac{CD}{CE}\), then \(CD = CE\times\sin E\).
\(CD=\sqrt{78}\times\sin(59^{\circ})\approx8.83\times0.857\approx7.6\)

Step3: Find \(DE\)

We know that \(\cos E=\frac{DE}{CE}\). Since \(\cos(59^{\circ})\approx0.515\) and \(CE = \sqrt{78}\approx8.83\)
\(DE=CE\times\cos E\).
\(DE=\sqrt{78}\times\cos(59^{\circ})\approx8.83\times0.515\approx4.5\)

Answer:

\(m\angle E = 59^{\circ}\), \(CD\approx7.6\), \(DE\approx4.5\)