Assumed knowledge The material in this module is a continuation of the module, Parallelograms and Rectangles, which is assumed knowledge for the present module. Thus the present module assumes:
Motivation Logical argument, precise definitions and clear proofs are essential if one is to understand mathematics. These analytic skills can be transferred to many areas in commerce, engineering, science and medicine but most of us first meet them in high school mathematics. Apart from some number theory results such as the existence of an infinite number of primes and the Fundamental Theorem of Arithmetic, most of the theorems students meet are in geometry starting with Pythagoras’ theorem. Many of the key methods of proof such as proof by contradiction and the difference between a theorem and its converse arise in elementary geometry. As in the module, Parallelograms and Rectangles, this module first stresses precise definitions of each special quadrilateral, then develops some of its properties, and then reverses the process, examining whether these properties can be used as tests for that particular special quadrilateral. We have seen that a test for a special quadrilateral is usually the converse of a property. For example, a typical property−test pair from the previous module is the pair of converse statements:
Congruence is again the basis of most arguments concerning rhombuses, squares, kites and trapezia, because the diagonals dissect each figure into triangles. A number of the theorems proved in this module rely on one or more of the previous theorems in the module. This means that the reader must understand a whole ‘sequence of theorems’ to achieve some results. This is typical of more advanced mathematics. In addition, two other matters are covered in these notes.
Content Symmetries of triangles, parallelograms and rectangles We begin by relating the reflection and rotation symmetries of isosceles triangles, parallelograms and rectangles to the results that we proved in the previous module, Paralleograms and Rectangles. The axis of symmetry of an isosceles triangle
This congruence result, however, establishes much more than the equality of the base angles. It also establishes that the angle bisector AM is the perpendicular bisector of the base BC. Moreover, this fact means that AM is an axis of symmetry of the isosceles triangle. These basic facts of about isosceles triangles will be used later in this module and in the module, Circle Geometry: Theorem
This line is an axis of symmetry of the isosceles triangle. It has, as a consequence, the interesting property that the centroid, the incentre, the circumcentre and the orthocentre of Extension − Some further tests for a triangle to be isosceles The theorem above suggests three possible tests for a triangle to be isosceles. The first two are easy to prove, but the third is rather difficult because simple congruence cannot be used in this ‘non-included angle’ situation.
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EXERCISE 1 Use congruence to prove that b the line through A perpendicular to BC bisects c the angle bisector of [Hint: For part c, let Suppose by way of contradiction that AC < AB. Choose P on the interval AB so that AP = AC, and join PM. The symmetries of an equilateral triangle An equilateral triangle is an isosceles triangle in three different ways, so the three vertex angle bisectors form three axes of symmetry meeting each other at 60°. In an equilateral triangle, each vertex angle bisector is the perpendicular bisector of the opposite side − we proved in the previous module that in any triangle, these three perpendicular bisectors are concurrent. They meet at a point which is the centre of a circle through all three vertices. The point is called the circumcentre and the circle is called the circumcircle of the triangle. An equilateral triangle is also congruent to itself in two other orientations:
corresponding to the fact that it has rotation symmetry of order 3. The centre of this rotation symmetry is the circumcentre O described above, because the vertices are equidistant from it. Other triangles do not have reflection or rotation symmetry In a non-trivial rotation symmetry, one side of a triangle is mapped to a second side, In a reflection symmetry, two sides are swapped, so the triangle must be isosceles. Thus a triangle that is not isosceles has neither reflection nor rotation symmetry. Such a triangle is called scalene. Rotation symmetry of a parallelogram
Reflection symmetry of a rectangle
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EXERCISE 2
Axes of symmetry of triangles, parallelograms and rectangles
Rhombuses The Greeks took the word rhombos from the shape of a piece of wood that was whirled about the head like a bullroarer in religious ceremonies. This derivation does not imply a definition, unlike the words ‘parallelogram’ and ‘rectangle’, but we shall take their classical definition of the rhombus as our definition because it is the one most usually adopted by modern authors. Definition of a rhombus A rhombus is a quadrilateral with all sides equal. First property of a rhombus − A rhombus is a parallelogram Since its opposite sides are equal, a rhombus is a parallelogram − this was our second test for a parallelogram in the previous module. A rhombus thus has all the properties of a parallelogram:
When drawing a rhombus, there are two helpful orientations that we can use, as illustrated below.
The rhombus on the left looks like a ‘pushed-over square’, and has the orientation we usually use for a parallelogram. The rhombus on the right has been rotated so that it looks like the diamond in a pack of cards. It is often useful to think of this as the standard shape of a rhombus. Constructing a rhombus using the definition It is very straightforward to construct a rhombus using the definition of a rhombus. Suppose that we want to construct a rhombus with side lengths 5cm and acute vertex angle 50°.
The figure OAPB is a rhombus because all its sides are 5cm.
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EXERCISE 3 Use the cosine rule (or drop a perpendicular and use simple trigonometry) to find the lengths of the lengths of the diagonals of the rhombus OAPB constructed above.
Then PQ || Second property of a rhombus − Each diagonal bisects two vertex angles Theorem Each diagonal of a rhombus bisects the vertex angles through which it passes.
That is AC bisects
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EXERCISE 4 Prove this result using the congruence The axes of symmetry of a rhombus
Thus the diagonals of a rhombus are axes of symmetry. The following property shows that these two axes are perpendicular. Third property of a rhombus − The diagonals are perpendicular The proof given here uses the theorem about the axis of symmetry of an isosceles triangle proven at the start of this module. Two other proofs are outlined as exercises. Theorem
Proof Let ABCD be a rhombus, To prove that AC ⊥ BD. By the previous theorem, AM is the angle bisector of Hence AM ⊥ BD, because A is the apex of the isosceles triangle ABD, The diagonals also bisect each other because a rhombus is a parallelogram, so we usually state the property as ‘The diagonals of a rhombus bisect each other at right angles.’
click for screencast We now turn to tests for a quadrilateral to be a rhombus. This is a matter of establishing that a property, or a combination of properties, gives us enough information for us to conclude that such a quadrilateral is a rhombus. First test for a rhombus − A parallelogram with two adjacent sides equal
Theorem If two adjacent sides of a parallelogram are equal, then it is a rhombus. This test is often taken as the definition of a rhombus. Second test for a rhombus − A quadrilateral whose diagonals bisect each other Theorem A quadrilateral whose diagonals bisect each other at right angles is a rhombus. Proof
We prove that DA = AB. It follows similarly that AB = BC and BC = CD
So AB = AD and by the first test above ABCD is a rhombus. A quadrilateral whose diagonals bisect each other is a parallelogram, so this test is often stated as ‘If the diagonals of a parallelogram are perpendicular, then it is a rhombus.’ This test gives us another construction of a rhombus.
This figure is a rhombus because its diagonals bisect each other at right angles.
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EXERCISE 6 If the circles in the constructions above have radius 4cm and 6cm, what will the side length and the vertex angles of the resulting rhombus be? Third test for a rhombus − A quadrilateral in which the diagonals bisect Theorem If each diagonal of a quadrilateral bisects the vertex angles through which it passes, then the quadrilateral is a rhombus. Proof
To prove that ABCD is a rhombus.
so ABCD is a rhombus because it is a parallelogram with a pair of adjacent sides equal. Extension − Quadrilaterals whose diagonals are perpendicular The converse of a property is not necessarily a test. For example, a quadrilateral with perpendicular diagonals need not be a rhombus − just place two sticks across each other at right angles and join their endpoints. The following exercise gives an interesting characterisation of quadrilaterals with perpendicular diagonals. Both parts of the proof are applications of Pythagoras’ theorem. One half is straightforward, the other requires proof by contradiction and an ingenious construction.
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EXERCISE 7 Use Pythagoras’ theorem to prove that the diagonals of a convex quadrilateral are perpendicular if and only if the sum of the squares of each pair of opposite sides are equal. Squares We usually think of a square as a quadrilateral with all sides equal and all angles right angles. Now that we have dealt with the rectangle and the rhombus, we can define a square concisely as: Definition of a square A square is a quadrilateral that is both a rectangle and a rhombus. Properties of a square A square thus has all the properties of a rectangle, and all the properties of a rhombus.
Symmetries of a square The intersection of the two diagonals is the circumcentre of the circumcircle through all four vertices. We have already seen, in the discussion of the symmetries of a rectangle, that all four axes of symmetry meet at the circumcentre.
ABCD corresponding to the fact that it has rotation symmetry of order 4. The centre of the rotation symmetry is the circumcentre, because the vertices are equidistant from it. Constructing a square
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EXERCISE 8 What radius should the circle have for the second construction above to produce a square of side length 6cm? Extension − A dissection problem The following problem requires the construction that divides a given interval in a given ratio − see the module Constructions.
click for screencast Kites Some of the distinctive properties of the diagonals of a rhombus hold also in a kite, which is a more general figure. Because of this, several important constructions are better understood in terms of kites than in terms of rhombuses. Definition of a kite A kite is a quadrilateral with two pairs of adjacent equal sides.
A kite may be convex or non-convex, as shown in the diagrams above. Only the convex cases are presented in the proofs below − the non-convex cases are similar, but are left Constructing a kite using its definition The definition allows a straightforward construction using compasses. Suppose that we want to construct a kite with side lengths 8cm and 5cm, with the two 8cm sides meeting at 60°.
The last two circles meet at two points P and P0, one inside the large circle and one outside, giving a convex kite and a non-convex kite meeting the specifications. Notice that the reflex angle of a non-convex kite is formed between the two shorter sides.
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EXERCISE 10 What will the vertex angles and the lengths of the diagonals be in the kites constructed above? First property of a kite − The axis of symmetry Theorem
a bThe diagonal AC is an axis of reflection symmetry of the kite. c The axis AC bisects the vertex angles at A and C. d Proof The congruence follows from the definition, and the other parts follow from the congruence. Second property of a kite − The axis is the perpendicular bisector of the other diagonal Theorem The axis of a kite is the perpendicular bisector of the other diagonal. Proof Let ABCD be a kite with AB = AD and CB = CD. and let the diagonals of the kite meet at M.
Hence BM = MD and AM ⊥ BD So AC ⊥ BD
click for screencast Tests for a kite The converses of some these properties of a kite are tests for a quadrilateral to be a kite. Theorem
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EXERCISE 12 Prove this result using the given diagram. Theorem
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EXERCISE 13 Prove this result using the given diagram.
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EXERCISE 14 Is it true that if a quadrilateral has a pair of opposite angles equal and a pair of adjacent sides equal, then it is a kite? Kites and geometric constructions Three of the most common ruler-and-compasses constructions can be explained in terms of kites.
Notice that the radii of the arcs meeting at P need not be the same as the radius of the first arc with centre O. Notice that the radii of the arcs meeting at Q need not be the same as the radii of the original arc with centre P.
In the diagram to the left, the radii of the arcs meeting at P are not the same as the radii of the arcs meeting at Q. Of course it is usual in this construction, and far more convenient, to use equal radii − as in the diagram to the right − in which case the figure constructed is a rhombus. Trapezia Trapezia also have a characteristic property involving the diagonals, but the property concerns areas, not lengths or angles. Definition of a trapezium
Although the name is Latin (the plural is ‘trapezia’), it originally comes from the Greek word trapeza, meaning ‘table’. The figure is called a ‘trapezoid’ in the USA. The angles of a trapezium
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EXERCISE 15 What sort of figure is both a kite and a trapezium? The diagonals of a trapezium The diagonals of a convex quadrilateral dissect the quadrilateral into four triangular regions, as shown in the diagrams below. In a trapezium, two of these triangles have the same area, and the converse of this property is a test for a quadrilateral to be a trapezium. These results are written as exercises because they are not usually regarded as standard theorems for students to know.
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EXERCISE 16 Prove that
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EXERCISE 17 Extension − Isosceles trapezia The trapezia that occur in this exercise are called isosceles trapezia. Further results about isosceles trapezia can be found at https://en.wikipedia.org/wiki/Isosceles_trapezoid.
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EXERCISE 18 b Prove that AB = DC if and only if the line through the midpoints F of AD and G of BC is an axis of symmetry of the trapezium. c Prove that no other line is an axis of symmetry of the isosceles trapezium. Links forward This module completes the study of special quadrilaterals using congruence. Similarity is a generalisation of congruence, and when it has been developed, some further results about special quadrilaterals will become possible. The module, Circle Geometry will use some special quadrilaterals, and will also introduce cyclic quadrilaterals, which are quadrilaterals whose four vertices all lie on the one circle − they will be the last special quadrilaterals discussed in these modules. All triangles have both a circumcircle and an incircle. The only quadrilaterals that have a circumcircle are those with opposite angles supplementary, the situation with incircles is interesting. For example, a rhombus always has an incircle.
As an easy exercise show that if the lengths of the diagonals of the rhombus are p and q and the radius of the incircle is r then
A kite has an incircle as well but its radius is difficult to calculate. A kite is determined by the triangle with side lengths a, b and included angle θ. If the lengths of the diagonals are p and q show that : 2pq = ab sin θ
When complex numbers are graphed on Argand diagrams, many arithmetic and algebraic results are proved or illustrated using special quadrilaterals. In particular if z1 and z2 are two complex numbers of equal modulus then the four numbers, z1, z1+ z2 and z2 form a rhombus so, as a consequence, z1 + z2 and z1,− z2 are perpendicular vectors. History and applications This illustrates very well the constant attitude in mathematics that an investigation is not complete until a theorem with a true converse has been identified. It reminds us too that logic, accompanied by the intuition of diagrams, should always be a strong motivation in geometry. Whenever a surface is divided up, triangles and special quadrilaterals are involved, particularly when parallel lines are used in the dissection. Thus surveyors analysing suburban blocks or farming lots will try to use the simplest geometric shapes in their analysis, and architects, who often have great freedom to invent striking patterns for their building, often use special quadrilaterals other than simple squares and rectangles in their designs. Infinite tilings of the plane, for example, are possible with any other quadrilateral. Trigonometry is also an essential part of these processes, and trigonometry and geometry should be seen as a unit rather than as two disconnected topics. Several exercises in these modules have required such connections to be made. A standard problem for computer programmers is to encrypt pictures with as little storage as possible, and they typically divide up the picture into simple geometrical shapes as part of that process − a recent study uses interlocking trapezia for this purpose. All the special quadrilaterals of this and the previous module, apart from the kite, were studied by the ancient Greeks as part of their systematic investigation of geometry. The kite was named and brought to attention in modern times, partly because that it clarifies several important geometric constructions, but also because it demonstrates that some of the properties of a rhombus hold in more general quadrilaterals. ANSWERS TO EXERCISES EXERCISE 1
c Suppose AC < AB. Choose P on the internal AB so that AP = AC and join PM
EXERCISE 2 In the triangles APM and CQM in the given diagram:
EXERCISE 3
EXERCISE 4 The congruence EXERCISE 5 a
or A rhombus is a parallelogram. So BM = MD. Hence EXERCISE 6 Using Pythagoras’ theorem, the side length is 2 EXERCISE 7 First, let ABCD be a convex quadrilateral whose diagonals meet at right angles at M.
Conversely, let ABCD be a convex quadrilateral in which the diagonals are not perpendicular. The diagram will be as drawn on the right or its reflection, and it will be sufficient to consider only the one case. Let the lengths be as given on the figure, where
EXERCISE 8 Using Pythagoras’ theorem, the required radius is 3 EXERCISE 9
BP : PC = DQ : QC = 2 : 1 Then the triangles APB and APC have the same altitude AB, and their bases are in the ratio 2 : 1, so area Similarly area or show that the area of each region is b A similar argument works EXERCISE 10 Take the convex case first. Using equilateral triangles and Pythagoras’ theorem, the axis is 4 In the non-convex case, the axis is 4 EXERCISE 11 a Since the axis AC bisects the vertex angle at A,
from which it follows that
EXERCISE 12 Using the AAS congruence test, So AB = AD, BC = DC and the quadrilateral is a kite. EXERCISE 13 Using the SAS congruence test, So AB = AD and CB = CD and the quadrilateral is a kite. EXERCISE 14 Diagram: Kite 10
If one of the equal angles is included by the given sides, then there is no reason for the figure to be a kite, as is illustrated in the diagram on the left above. If neither equal angle is included by the equal sides, then the figure is a kite. Be careful, however, because joining the diagonal AC in the diagram on the right above would not give a congruence proof because the angle in each triangle would be non-included. Instead, join the other diagonal BD, as in the diagram on the right above.
EXERCISE 15
so ABCD is a rhombus, because it is a parallelogram with two adjacent sides equal. EXERCISE 16 The triangles Subtracting the triangle MBC from both regions, area EXERCISE 17 Adding area These two triangles have the same base BC, so they have the same perpendicular height. Hence AD || BC. EXERCISE 18
Conversely, suppose that AB = DC. Then Drop perpendiculars AL and DM to the line BC.
so reflection in FG swaps B and C, and swaps A and D, as required. c Since AD ≠ BC, a reflection cannot swap AD and BC. If a reflection swapped AD and AB, then it would also swap BC and DC, so ABCD would be a kite with parallel sides, so it would be a parallelogram. Similarly a reflection cannot swap AD and DC. The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. |
In the parallelogram abcd , angle a is two thirds of angle b, the measure of angle is d is _________
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