Quadrilaterals | IT

Question 2

Are there other ways to define a rectangle?

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Solution

A rectangle can be defined in multiple ways, all describing the same geometric shape.

Step 1 — Rectangle as a special type of parallelogram

Let us first remember what a parallelogram is. A parallelogram is a quadrilateral where opposite sides are parallel. Its opposite angles are equal. Its consecutive angles add up to 180 degrees.

A rectangle is a parallelogram where all its interior angles are right angles. This means each of its four angles measures 90 degrees.

Step 2 — Parallelogram with just one right angle

Let us consider a parallelogram. If even one of its interior angles is a right angle (90 degrees). We know that consecutive angles in a parallelogram are supplementary. This means they add up to 180 degrees. So, if one angle, say angle A, is 90 degrees. Then the angle next to it, angle B, must be 180∘−90∘=90∘180^\circ - 90^\circ = 90^\circ. Also, opposite angles in a parallelogram are equal. So, if angle A is 90 degrees, then angle C (opposite to A) is also 90 degrees. And if angle B is 90 degrees, then angle D (opposite to B) is also 90 degrees. This shows that all four angles become 90 degrees. Therefore, a parallelogram with at least one right angle is a rectangle.

Step 3 — Quadrilateral with special diagonals

Let us consider any quadrilateral. If its two diagonals are equal in length. And if these diagonals bisect each other (cut each other into two equal parts). When the diagonals of a quadrilateral bisect each other, the quadrilateral is always a parallelogram. Now we have a parallelogram where the diagonals are equal. Let the parallelogram be ABCD, with diagonals AC and BD. If diagonal AC has the same length as diagonal BD. Let us look at two triangles: triangle DAB and triangle CDA. Side AD is common to both triangles. Side AB is equal to side CD (because opposite sides of a parallelogram are equal). Diagonal BD is equal to diagonal AC (this is our given condition). So, triangle DAB is congruent to triangle CDA (by the SSS congruence rule). This means their corresponding angles are equal. So, angle DAB must be equal to angle CDA. We also know that consecutive angles in a parallelogram add up to 180 degrees. So, angle DAB + angle CDA = 180 degrees. Since angle DAB = angle CDA, each must be 180∘/2=90∘180^\circ / 2 = 90^\circ. As we saw in Step 2, if one angle of a parallelogram is 90 degrees, all its angles are 90 degrees. So, a quadrilateral whose diagonals are equal and bisect each other is a rectangle.

Diagram 1

Step 4 — Quadrilateral with opposite sides equal and one right angle

Let us consider a quadrilateral. If its opposite sides are equal in length. This property means that the quadrilateral is a parallelogram. Now we have a parallelogram that also has one right angle. This is the exact situation we explained in Step 2. If a parallelogram has one right angle, then all its angles become 90 degrees. So, a quadrilateral with opposite sides equal and one right angle is a rectangle.

Answer

Yes. There are other ways to define a rectangle, even though they must mean the same thing.

(i) Definition (using parallelogram idea): A rectangle is a parallelogram whose all angles are right angles. (ii) Definition (using one right angle): A parallelogram with even one right angle is a rectangle (because if one angle is 90°, all others automatically become 90°). (iii) Definition (using diagonals): A quadrilateral whose diagonals are equal and bisect each other is a rectangle. (iv) Definition (opposite sides equal + right angle): A quadrilateral with opposite sides equal and one right angle is a rectangle.

More questions in IT

Q1

Observe the following figures.

Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?

Q2

Are there other ways to define a rectangle?

Q3

A Carpenter's Problem

A carpenter needs to put together two thin strips of wood, as shown in Fig. 1, so that when a thread is passed through their endpoints, it forms a rectangle. She already has one 8 cm long strip. What should be the length of the other strip? Where should they both be joined?

Let us first model the structure that the carpenter has to make. The strips can be modelled as line segments. They are the diagonals of the quadrilateral formed by their endpoints. For the quadrilateral to be a rectangle, we need to answer the following questions —

  1. What is the length of the other diagonal?
  2. What is the point of intersection of the two diagonals?
  3. What should the angle be between the diagonals?
Q4

Can the following equalities be used to establish that ΔAOD≅ΔCOB\Delta AOD \cong \Delta COB?

  • AO=COAO = CO (proved above)
  • ∠AOB=∠COD\angle AOB = \angle COD (vertically opposite angles)
  • AD=CBAD = CB
Q5

Context: Let us check what quadrilateral we get if we draw the two diagonals such that their lengths are equal, they bisect each other and have an arbitrary angle, say 60∘60^\circ, between them as shown in the figure to the right.

Q. Can you find all the remaining angles?

Q6

Context: In ΔAOB\Delta AOB, since OA=OBOA = OB, the angles opposite them are equal, say aa.

Q. Can you find the value of aa?

Q7

Can we now identify what type of quadrilateral ABCD is?

Notice that its angles all add up to 90° (30° + 60°).

Q8

What can we say about its sides?

Q9

Will ABCD remain a rectangle if the angles between the diagonals are changed? Can we generalise this?

Take one of the angles between the diagonals as xx.

Q10

Context: We can compute the four angles between the diagonals to be x,x,180−x,x, x, 180 - x, and 180−x.180 - x.

Q. Can you find the other angles?

Q11

Context: Since we know that ΔAOB\Delta AOB is isosceles, we can denote the measures of both of its base angles by aa.

Q. What is the value of aa (in degrees) in terms of xx?

Q12

Context: Thus, all four angles of the quadrilateral ABCD are 90°.

Q. What can we say about AB and CD, and AD and BC?

Q13

In the earlier definition, we stated that a rectangle has (a) opposite sides of equal length, and (b) all angles equal to 90°. Would we be wrong if we just define a rectangle as a quadrilateral in which all the angles are 90°?

Q14

If you think that this definition is incomplete, try constructing a quadrilateral in which the angles are all 90° but the opposite sides are not equal.

Are you able to construct such a quadrilateral?

Q15

Is it wrong to write ΔBAD ≅ ΔCDB? Why?

Q16

Can you similarly show that AB is parallel to DC (AB || DC)?

Q17

In the quadrilaterals below, are there any non-rectangles?

Q18

Let us consider the Carpenter's Problem again. If the wooden strips have to be placed such that the thread passing through their endpoints forms a square, what must be done?

Q19

What more needs to be done to get equal sidelengths as well? Can this be achieved by properly choosing the angle between the diagonals? See if you can reason and/or experiment to figure this out!

Q20

Can this be used to find the angles ∠BOA\angle\text{BOA} and ∠BOC\angle\text{BOC} formed by the diagonals?

Q21

Context: The diagonals of a square are of equal lengths and bisect each other at right angles.

Q. Using this fact, construct a square with a diagonal of length 8 cm.

Q22

Context: Since a square is a special type of rectangle, all the properties of a rectangle hold true for a square.

Q. Verify if this is true by going through geometric reasoning in Deduction 1 and Deduction 2, and see if they apply to a square as well.

Q23

Q. Similarly, find ∠2\angle 2 and ∠4\angle 4.

Q24

4.2 Angles in a Quadrilateral

Is it possible to construct a quadrilateral with three angles equal to 90° and the fourth angle not equal to 90°?

Q25

But why not?

Q26

Are there quadrilaterals that have parallel opposite sides that are not rectangles?

Q27

Construct such a figure by recalling how parallel lines can be constructed using a ruler and a set-square, or a compass and a ruler.

Q28

Context: Consider a parallelogram ABCDABCD with adjacent sides of lengths 4 cm4\text{ cm} and 5 cm5\text{ cm}, and an angle of 30∘30^\circ between them.

Q. What are the remaining angles of the parallelogram? What are the lengths of the remaining sides? See if you can reason out and/or experiment to figure these out.

Q29

Deduction 7— What can we say about the sides of a parallelogram?

By looking at a parallelogram, it appears that the opposite sides are equal. Can we again use congruence to show this? Which two triangles can be considered for this?

Q30

Is it wrong to write ΔABD≅ΔCBD\Delta\text{ABD} \cong \Delta\text{CBD}? Why?

Q31

Are the diagonals of a parallelogram always equal? Check with the parallelogram that you have constructed.

Q32

Context: We see that the diagonals of a parallelogram need not be equal.

Q. Do they bisect each other (do they intersect at their midpoints)? Reason and/or experiment to figure this out.

Q33

Is it wrong to write ΔAOE≅ΔSOY\Delta\text{AOE} \cong \Delta\text{SOY}? Why?

Q34

Do the diagonals of a parallelogram intersect at a particular angle?

Q35

What are the other angles of the rhombus ABCD that we have constructed? Reason and/or experiment to figure this out.

Q36

It can be seen that ΔGAE≅ΔMAE\Delta GAE \cong \Delta MAE (How?)

Q37

So a rhombus is a parallelogram, and a rectangle is also a parallelogram. How can this be represented using a Venn diagram?

Q38

Where will the set of squares occur in this diagram?

Q39

Are the diagonals of a rhombus equal?

Q40

Do the diagonals of a rhombus intersect at any particular angle? Reason out and/or experiment to figure this out!

Q41

In the rhombus GAME, we have ΔGEO≅ΔMEO\Delta\text{GEO} \cong \Delta\text{MEO} (why?).

Q42

In the kite, show that the diagonal BDBD

(i) bisects ∠ABC\angle ABC and ∠ADC\angle ADC,

(ii) bisects the diagonal ACAC, that is, AO=OCAO = OC, and is perpendicular to it.

Hint: Is ΔAOB≅ΔCOB\Delta AOB \cong \Delta COB?

Q43

Construct a trapezium. Measure the base angles (marked in the figure).

Q44

Can you find the remaining angles without measuring them?

Q45

How do we construct an isosceles trapezium?

Q46

Construct an isosceles trapezium UVWX, with UV || XW. Measure ∠U.

Q47

Now, it can be shown that ΔUXY≅ΔVWZ\Delta\text{UXY} \cong \Delta\text{VWZ}. (How?)

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