Question

ex 6: solve the following triangle.

Explanation:

Step1: Find angle D

The sum of angles in a triangle is 180°. Given one - angle is 90° (at F) and another is 25° (at E). So, $\angle D=180^{\circ}-90^{\circ}-25^{\circ}=65^{\circ}$.

Step2: Use the sine - rule to find side EF

The sine - rule states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ in a triangle. Let $EF = x$, $DF = 5.0$ cm, $\angle D = 65^{\circ}$, and $\angle E=25^{\circ}$. Then $\frac{x}{\sin D}=\frac{DF}{\sin E}$. Substituting the values, we have $x=\frac{DF\times\sin D}{\sin E}$. Since $\sin D=\sin65^{\circ}\approx0.9063$, $\sin E=\sin25^{\circ}\approx0.4226$, and $DF = 5.0$ cm, $x=\frac{5\times0.9063}{0.4226}\approx10.7$ cm.

Step3: Use the cosine - rule or Pythagorean theorem to find side ED

Using the Pythagorean theorem in right - triangle DEF ($ED^{2}=EF^{2}+DF^{2}$). Substituting $EF\approx10.7$ cm and $DF = 5.0$ cm, $ED=\sqrt{10.7^{2}+5.0^{2}}=\sqrt{114.49 + 25}=\sqrt{139.49}\approx11.8$ cm.

Answer:

$\angle D = 65^{\circ}$, $EF\approx10.7$ cm, $ED\approx11.8$ cm