Practical Geometry

A triangle can be constructed uniquely when three suitable measurements are known, but a quadrilateral needs five. In this chapter you will learn why: four sides alone do not fix the shape of a quadrilateral, since a square of side 4 cm can be pushed into a rhombus of the same side, but adding a diagonal or an angle makes the figure rigid. You will construct quadrilaterals using ruler, compasses and protractor when you know (i) four sides and one diagonal, (ii) two diagonals and three sides, (iii) two adjacent sides and three angles, and (iv) three sides and two included angles. In every case you will first draw a rough sketch, mark the given measurements and decide which triangle to construct first, checking that the triangle inequality and the angle sum of 360 degrees are satisfied. Finally, you will use the special properties of rhombuses, rectangles, squares, parallelograms and kites to construct them with fewer measurements.

Detailed Worksheet: Practical Geometry Section A - Definitions (10 marks) 1. How many independent measurements are needed to construct a unique quadrilateral? Why are fewer measurements needed for a triangle? (2 marks) 2. Explain, with a sketch of a square and a rhombus of the same side, why four sides alone do not fix a quadrilateral. (2 marks) 3. State two properties of the diagonals of a rhombus that are used in its construction. (2 marks) 4. Why is one measurement enough to construct a square, while two are needed for a rectangle? (2 marks) 5. Explain the terms diagonal, adjacent sides and included angle of a quadrilateral, using a sketch of quadrilateral ABCD. (2 marks) Section B - Calculations and Applications (15 marks) 6. Quadrilateral PQRS is to be constructed with PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR = 7 cm. Which two triangles will you construct and in which order? Check, using the triangle inequality, that both triangles are possible. (3 marks) 7. The diagonals of a rhombus are 6 cm and 8 cm. Using the fact that the diagonals bisect each other at right angles, find the length of each side. Write the steps of construction. (3 marks) 8. A student wants to construct ABCD with AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and diagonal AC = 11 cm. Show that the construction is impossible, and find the range of possible lengths of AC for which both triangles can be drawn. (3 marks) 9. In quadrilateral ABCD, angle A = 75 degrees, angle B = 105 degrees and angle C = 120 degrees. Find angle D. Would a quadrilateral with angles A = 100, B = 120 and C = 150 degrees be possible? Explain. (3 marks) 10. A rectangle has sides 5 cm and 4 cm. Find the length of its diagonal to one decimal place, and explain why the two diagonals of the rectangle will be equal in your construction. (3 marks) Section C - Diagrams (10 marks) 11. Construct quadrilateral ABCD with AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and AC = 7 cm. Show all construction arcs and write the steps. (4 marks) 12. Construct a rhombus whose diagonals are 5.2 cm and 6.4 cm. Measure and record its side. (3 marks) 13. Construct a square READ with RE = 5.1 cm. Measure both diagonals and state what you observe. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Ravi is asked to construct quadrilateral ABCD with AB = 4 cm, BC = 5 cm, angle A = 100 degrees, angle B = 120 degrees and angle C = 150 degrees. Explain why he cannot succeed. Then explain why four sides of 4 cm each can give more than one quadrilateral, and state one extra measurement that would make the figure unique. (5 marks) 15. A kite ABCD has AB = AD = 4 cm, CB = CD = 6 cm and diagonal BD = 5 cm. Draw a rough sketch, write the steps of construction, construct the kite and measure the diagonal AC. Explain why AC is perpendicular to BD. (5 marks) 16. A parallelogram has adjacent sides 4 cm and 6 cm and one angle of 60 degrees. Explain why only these three measurements are enough, find all four angles, and construct the parallelogram. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use a sharp pencil, ruler, compasses and protractor, draw a rough sketch before each construction and do not erase construction arcs.