Test questions for solving mixed general triangle problems

📚 Quick Study Guide

• 📐 Law of Sines: Useful when you know two angles and a side, or two sides and an angle opposite one of them. Formula: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• 📏 Law of Cosines: Use this when you know three sides, or two sides and the included angle. Formulas: $a^2 = b^2 + c^2 - 2bc \cos A$, $b^2 = a^2 + c^2 - 2ac \cos B$, $c^2 = a^2 + b^2 - 2ab \cos C$
• 🧭 Area of a Triangle: When you know two sides and the included angle, use the formula: $Area = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ac \sin B$. If you know all three sides, use Heron's formula: $Area = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ (semi-perimeter).
• 💡 Angle Sum: Remember that the sum of angles in any triangle is always 180 degrees: $A + B + C = 180^{\circ}$
• 🧮 Solving Strategy: Always visualize the problem. Draw a diagram. Identify what you're given and what you need to find. Choose the appropriate law (Sines or Cosines) based on the given information.

Practice Quiz

1. In triangle ABC, angle A = 45°, angle B = 60°, and side a = 10. Find side b.

   1. 12.25
   2. 13.66
   3. 11.55
   4. 9.89

2. In triangle XYZ, x = 7, y = 8, and z = 9. Find angle Z.

   1. 48.19°
   2. 81.79°
   3. 62.43°
   4. 74.33°

3. In triangle PQR, p = 15, q = 20, and angle R = 30°. Find the area of the triangle.

   1. 75
   2. 100
   3. 150
   4. 200

4. In triangle DEF, angle D = 120°, e = 10, and f = 12. Find side d.

   1. 19.08
   2. 20.50
   3. 17.46
   4. 22.11

5. In triangle LMN, angle L = 35°, angle M = 85°, and side n = 8. Find side l.

   1. 4.56
   2. 5.12
   3. 6.23
   4. 7.88

6. In triangle STU, s = 6, t = 10, and u = 12. Find angle S.

   1. 28.96°
   2. 30.00°
   3. 26.38°
   4. 32.20°

7. In triangle ABC, a = 13, b = 14, and c = 15. Find the area of the triangle.

   1. 78
   2. 84
   3. 90
   4. 96