Worksheet Chapter 9: Solution of Triangles

1. In triangle ABC, side a = 8.5 cm, angle A = 42°, and angle B = 68°. (a) Find the measure of angle C. (b) Calculate the length of side b.

2. A surveyor measures two points A and B from point O. OA = 120 m, OB = 150 m, and angle AOB = 75°. (a) Calculate the distance between A and B. (b) Calculate the shortest distance from point O to the line AB.

3. In triangle XYZ, XY = 12 cm, XZ = 15 cm, and the area of the triangle is 54 cm². (a) Find the two possible values for angle YXZ. (b) Using the obtuse angle YXZ, calculate the length of side YZ.

4. In a three-dimensional pyramid VABC with a horizontal triangular base ABC, AB = 8 cm, BC = 6 cm, and angle ABC = 90°. The vertex V is vertically above point B with VB = 10 cm. (a) Find the length of VA and VC. (b) Calculate the area of the inclined triangle VAC.

5. A triangular garden has sides of lengths 13 m, 14 m, and 15 m. (a) Find the semi-perimeter s of the garden. (b) Calculate the area of the garden using Heron's Formula.

6. In triangle PQR, p = 6 cm, q = 9 cm, and angle P = 32°. (a) Show that an ambiguous case exists for angle Q. (b) Calculate the two possible values for angle Q.

7. Diagram shows a quadrilateral ABCD where AB = 6 cm, BC = 8 cm, CD = 7 cm, AD = 11 cm, and angle ABC = 90°. (a) Calculate the length of diagonal AC. (b) Find angle ADC.

8. True or False: The Cosine Rule can be used to find the angles of a triangle if only the lengths of all three sides are given.

9. Diagram shows a quadrilateral PQRS. Given PQ = 9 cm, QR = 12 cm, RS = 10 cm, angle PQR = 60°, and angle PRS = 40°. (a) Calculate the length of PR. (b) Calculate angle QPR. (c) Find the length of PS. (d) Calculate the total area of quadrilateral PQRS.

10. In triangle ABC, AB = 7 cm, BC = 5 cm, and AC = 9 cm. (a) Find the size of the largest angle in triangle ABC. (b) Calculate the area of triangle ABC.