Mathematics Year 7: The diagonals of a quadrilateral and their properties
Mathematics, Year 7 (Portugal): What a diagonal is and what each kind of quadrilateral does with its two: bisect each other, be equal, cross at a right angle.
Two diagonals, three questions
A diagonal of a quadrilateral is a segment joining two vertices that are not consecutive; a quadrilateral has two. For each quadrilateral, ask three questions about them: do they cut each other in the middle? Are they equal? Do they cross at a right angle? Parallelogram: they bisect each other. Rectangle: they bisect each other and are equal. Rhombus: they bisect each other and are perpendicular. Square: all three. Kite (two pairs of equal consecutive sides): they are perpendicular, and one of them cuts the other in the middle.
Why diagonals help, and how to find their properties
Sides and angles are not always enough to recognise a quadrilateral, and diagonals give new clues: a square and a slanted rhombus both have four equal sides, but only in the square are the diagonals equal. To find these properties, build several quadrilaterals in dynamic geometry or by folding paper, measure, and record the results in a table. Spot the pattern and state a conjecture. A new case can support it, but a single counterexample knocks it down.
A case: a rectangle and an equation
A rectangle ABCD has diagonal AC = 3x + 2 and diagonal BD = 5x − 8 (in cm). The diagonals of a rectangle are equal, so 3x + 2 = 5x − 8. Then 2 + 8 = 5x − 3x, so 10 = 2x and x = 5. AC = 3 × 5 + 2 = 17 cm and BD = 5 × 5 − 8 = 17 cm: equal, as they should be. The diagonals of a parallelogram bisect each other, so the point M where they cross is 17 ÷ 2 = 8.5 cm from each vertex.
The trap: one property is rarely enough
Two false ideas, both reasonable. First: «the diagonals of a rectangle are perpendicular». Draw a 6 cm by 3 cm rectangle and measure the angle: it is not a right angle. They would only be perpendicular in a square. Second: «if the diagonals are perpendicular, it is a rhombus». A kite has perpendicular diagonals too, but unless it is a rhombus only one of them is cut in the middle. One property alone almost never identifies a figure: put the clues together.
Where you see this
A paper kite has two sticks that are its diagonals: the crossbar is cut in the middle by the long spine, and they meet at a right angle. A carpenter who builds a frame with equal opposite sides measures both diagonals: if they are equal, the corners are right angles. In dynamic geometry you can drag a vertex and see which diagonal properties survive, and which ones vanish, when the figure changes.
Test what you learned
Which property of the diagonals is true for EVERY parallelogram?
- a) They are always equal
- b) They always cross at a right angle
- c) Each one cuts the other in the middle
- d) One is always double the other
Correct answer: c) Each one cuts the other in the middle — Right: the diagonals of a parallelogram bisect each other, so the same holds for the rectangle, the rhombus and the square.
In a rectangle ABCD, the diagonal AC = 2x + 5 and the diagonal BD = 4x − 9 (in cm). How long is the diagonal AC?
- a) 7 cm
- b) 38 cm
- c) 9.5 cm
- d) 19 cm
Correct answer: d) 19 cm — Right: the diagonals of a rectangle are equal, so 2x + 5 = 4x − 9, x = 7 and AC = 2 × 7 + 5 = 19 cm (BD = 4 × 7 − 9 = 19 cm too).
Marta measured the diagonals of a quadrilateral and found that they are perpendicular. What can she safely conclude?
- a) It is a rhombus
- b) Not yet: it could be a rhombus or a kite, for example; she needs another clue
- c) It is a rectangle
- d) It cannot be a square
Correct answer: b) Not yet: it could be a rhombus or a kite, for example; she needs another clue — Right: one property rarely identifies a figure. If the diagonals also bisect each other, it is a rhombus; if only one is bisected, a kite.
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