We have identified different types of quadrilaterals — squares, rectangles, parallelograms, rhombi, kites and trapezia. One can identify more types (e.g., we could create a category of quadrilaterals that have equal-length opposite sides).
Suppose we have identified a category of quadrilaterals called Q, and we have to construct a quadrilateral of this type. For this, we are to use two thin sticks, put them together as diagonals so that the quadrilateral obtained by joining their endpoints is of type Q (see the figure).
- (i) (a) Suppose Q satisfies the following property.
If a quadrilateral is of type Q, then it has equal-length diagonals.
Should the two sticks be of equal length? Why or why not? - (i) (b) Will it matter how the two sticks are put together?
- (ii) Instead of the property mentioned above, suppose Q satisfies the following property.
If a quadrilateral has equal diagonals, then it is of type Q.
What will be your answers to (a) and (b) now?
Step-by-step solution
Idea: The two properties are converses of each other. ‘Type Q ⇒ equal diagonals’ says equal diagonals are needed for Q. ‘Equal diagonals ⇒ type Q’ says equal diagonals are enough for Q.
(i) (a) Suppose Q satisfies the following property.
If a quadrilateral is of type Q, then it has equal-length diagonals.
Should the two sticks be of equal length? Why or why not?
- The property says every quadrilateral of type Q has equal diagonals. The sticks are the diagonals.½ mark
- If the sticks were unequal, the diagonals would be unequal, and then the quadrilateral could not be of type Q. So yes, the sticks must be of equal length.½ mark
(i) (b) Will it matter how the two sticks are put together?
- The property only says type Q ⇒ equal diagonals. It does not say that equal diagonals are enough to make a quadrilateral of type Q (that would be the converse).½ mark
- So it may matter. Q may need more: if Q were the rectangles, the sticks would also have to cross at their midpoints; if Q were the squares, they would also have to cross at right angles at their midpoints. Equal sticks put together carelessly might give, say, an isosceles trapezium instead.½ mark
(ii) Instead of the property mentioned above, suppose Q satisfies the following property.
If a quadrilateral has equal diagonals, then it is of type Q.
What will be your answers to (a) and (b) now?
- Now equal diagonals are enough for type Q.½ mark
- (a) Taking two equal sticks is a sure way to get type Q. But the property does not say every Q has equal diagonals, so we cannot say the sticks must be equal; some type-Q quadrilaterals might have unequal diagonals.½ mark
- (b) With equal sticks, it does not matter how they are put together: as long as they cross so that joining their ends gives a quadrilateral, its diagonals are equal.½ mark
- By the new property, any such quadrilateral is of type Q.½ mark
Answer to write in the exam
(i) (a)
Type Q ⇒ equal diagonals.
Sticks = diagonals; unequal sticks ⇒ not type Q.
∴ Yes, the sticks must be of equal length.
(i) (b)
Property gives only: type Q ⇒ equal diagonals (not the converse).
Equal diagonals may not be enough, e.g. Q = rectangles also needs the sticks to bisect each other.
∴ Yes, it may matter how they are put together.
(ii)
Equal diagonals ⇒ type Q.
(a) Equal sticks are enough to get type Q (not shown to be necessary).
(b) Any crossing of equal sticks gives equal diagonals ⇒ type Q.
∴ It does not matter how the equal sticks are put together.
Common mistakes that cost marks
- Treating the two properties as the same. The second is the converse of the first, so the answers change.
- Saying in (ii) that the sticks ‘must’ be equal. The new property only says equal diagonals are enough.
- Saying in (i)(b) that it never matters. The first property does not tell us equal diagonals are enough.
How this can come in the exam
Type R quadrilaterals satisfy: ‘If the diagonals of a quadrilateral bisect each other, then it is of type R.’ Sticks of 10 cm and 14 cm are crossed at their midpoints. The quadrilateral formed is:
- certainly of type R
- certainly not of type R
- of type R only if the sticks are equal
- of type R only if they cross at right angles
Show answer
(A) certainly of type R
Crossing at the midpoints makes the diagonals bisect each other, and that is enough for type R.
Type S quadrilaterals satisfy: ‘If a quadrilateral is of type S, then its diagonals bisect each other.’ Must the two sticks cross at their midpoints? Must they be of equal length?
Show answer
Yes, they must cross at their midpoints, because every type-S quadrilateral has diagonals that bisect each other (1 mark). The property says nothing about lengths, so it does not require equal sticks (1 mark).Try one yourself
Type T is defined by: ‘A quadrilateral is of type T exactly when its diagonals are equal and bisect each other.’ How should the sticks be chosen and put together?
Show answer
Two equal sticks, crossed at their midpoints (at any angle). These are exactly the rectangles.
More questions like this
- If a proposition is true, then is its converse always true?
- Statement 1: If two sides of a triangle are equal, then the angles opposite the equal sides are equal.
Statement 2: If two angles of a triangle are equal, then the sides opposite the equal angles have equal lengths.
We have proved the first statement in an earlier grade. Is the second statement true? Can you prove it?
(Hint: Draw the altitude from the vertex containing the third angle.) - Proposition P: If it rains, then the road is wet.
Converse Q: If the road is wet, then it has rained. - Proposition P: If a number is a multiple of 6, then it is a multiple of 3.
Converse Q: If a number is a multiple of 3, then it is a multiple of 6. - In this example, n is any positive integer.
Proposition P: If n is a perfect square, then it has an odd number of factors.
Converse Q: If n has an odd number of factors, then it is a perfect square.
You may recall that we came across these statements in the previous grade. Which of them are true?