MODULE - 9 QUADRILATERALS

Trough investigations produce conjectures and generalizations related to triangles, quadrilaterals and other polygons, and attempt to validate, justify, explain or prove them, using any logical method. (LO 3 AS 2)

DEFINITIONS OF QUADRILATERALS: A trapezium is a quadrilateral with one pair of opposite sides parallel.

2. A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

3. A rectangle is a parallelogram with all interior angles equal to 90°.

4. A rhombus is a parallelogram with equal sides

5. A square is a rectangle with equal sides.

6. A kite is a quadrilateral with two pairs of adjacent sides equal in length and opposite sides not equal in length.

LET’S INVESTIGATE!!!!! The exercise which follows will require you to work in groups of two or three. You will be required to explore the various properties of the quadrilaterals using Euclidean Geometry, Analytical Geometry or Transformation Geometry. The assessment task that follows will be a mind map on the properties of the quadrilaterals.

1. ABCD is a kite. Prove that: (a) (b) (c) (d)

ABCD is a kite with BE = ED (a) Why is (b) Why is (c) Prove that (d) Prove that

3. ABCD is a parallelogram. (a) Prove that (b) Show that (c) Show that

4. In the diagram that follows, a right-angled isosceles triangle is shown. Draw the image of this triangle if it is reflected about the dotted vertical line. Fill in the equal sides and angles of the two triangles. (b) Now reflect the newly formed triangle about the dotted horizontal line and draw the image. Fill in the equal sides and angles. (c) Now reflect the previously formed triangle about the vertical line and once again draw the image. Fill in the equal sides and angles. You should now have four triangles together forming a square.

Now write down as many properties of the square in terms of its sides, angles and diagonals.

5. In the diagram that follows, a right-angled scalene triangle is shown. Draw the image of this triangle if it is reflected about the dotted vertical line. Fill in the equal sides and angles of the two triangles. (b) Now reflect the newly formed triangle about the dotted horizontal line and draw the image. Fill in the equal sides and angles. (c) Now reflect the previously formed triangle about the vertical line and once again draw the image. You should now have four triangles together forming a rhombus.

6. In the diagram, ABCD is a parallelogram with its diagonals intersecting at E. Use analytical methods to answer the following questions.

(a) Calculate the gradient of AD and BC. What can you conclude? (b) Calculate the gradient of AB and DC. What can you conclude? (c) Calculate the length of AD and BC. What can you (d) Calculate the length of AB and DC. What can you (e) Calculate the coordinates of the midpoint of AC and the midpoint of BD. (f) Calculate the length of AE and EC. What can you (g) Calculate the length of BE and ED. What can you (h) Now write down as many properties of the parallelogram in terms of its sides and diagonals.

7. In the diagram, ABCD is a rectangle with its diagonals intersecting at E. Use analytical methods to answer the following questions.

(a) Calculate the length of AD and BC. What can you conclude? (b) Calculate the length of AB and DC. What can you (c) Calculate the coordinates of the midpoint of AC and the midpoint of BD. What can you conclude? (d) Calculate the length of AE and EC. What can you (e) Calculate the length of BE and ED. What can you (f) Calculate the length of AC and BD. What can you (g) Now write down as many properties of the rectangle in terms of its sides and diagonals.

8. ABCD is an isosceles trapezium (a) Why is AE = DF? (b) Prove, using congruency, that

ASSESSMENT TASK MIND MAP Create a mind map of the properties of the quadrilaterals in this module based on the information you obtained from the previous exercise. Use words and draw diagrams to illustrate the various properties in terms of the sides, angles and diagonals. Your mind map will be marked according to the following rubric:

Assessment Criteria No attempt at giving properties or all properties given are incorrect (0) Some of the properties are incorrect or some are not given (1) All properties are correct (2) Properties of an isosceles trapezium Properties of a parallelogram Properties of a rectangle Properties of a rhombus Properties of a square Properties of a kite Use of diagrams Use of words

Now answer the following questions: On your own set of axes, draw with vertices A (3; 2), B (1; 0) and D (5 ; 0). 2. Now draw , the image of under the transformation rule:

3. Calculate the length of AB, AD, DC and BC using the distance formula. 4. Calculate the gradient of AB and AD and hence show that AB AD. 5. Calculate the gradient of BC and DC and hence show that BC DC. 6. Show, by using gradients, that the opposite sides of figure ABCD are parallel. 7. Show that the diagonals of figure ABCD bisect each other at right angles. 8. What type of quadrilateral is ABCD? Provide detailed reasons using the information obtained from the previous questions.