Question

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

Explanation:

Step1: Find length of \(CD\) using Pythagorean theorem

In right - triangle \(BCD\), by Pythagorean theorem \(CD=\sqrt{BC^{2}+BD^{2}}\). Given \(BC = 4\) and \(BD=2\), then \(CD=\sqrt{4^{2}+2^{2}}=\sqrt{16 + 4}=\sqrt{20}\approx4.5\).

Step2: Find \(m\angle D\) using trigonometric ratio

\(\tan D=\frac{BC}{BD}\). Since \(BC = 4\) and \(BD = 2\), \(\tan D=\frac{4}{2}=2\). Then \(D=\arctan(2)\approx63.4^{\circ}\).

Step3: Find \(m\angle C\) using angle sum property of a triangle

In \(\triangle BCD\), \(\angle B = 90^{\circ}\). By angle sum property of a triangle (\(\angle B+\angle C+\angle D=180^{\circ}\)), \(\angle C=180^{\circ}-\angle B-\angle D\). Substituting \(\angle B = 90^{\circ}\) and \(\angle D\approx63.4^{\circ}\), we get \(\angle C=180^{\circ}-90^{\circ}-63.4^{\circ}=26.6^{\circ}\).

Answer:

\(m\angle D = 63.4^{\circ}\), \(m\angle C = 26.6^{\circ}\), \(CD = 4.5\)