Worksheet Chapter 9: Solution of Triangles

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.