Quadrilaterals | IT

Question 9

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 $x$ .

Solution

Understand the Question

A quadrilateral whose diagonals are equal in length and bisect each other is always a rectangle.
If the angle $x$ between the diagonals is changed while keeping the diagonals equal and mutually bisecting, the side lengths change, but the interior angles remain $90^\circ$ .
If the angle between the diagonals becomes $90^\circ$ , all four sides become equal, making it a square (which is a special rectangle).

Step 1 · Identify the Quadrilateral

From the figure, the diagonals $AC$ and $BD$ intersect at point $O$ such that $AO = BO = CO = DO$ .

Since $AO = OC$ and $BO = OD$ , the diagonals bisect each other.
Since $AC = AO + OC$ and $BD = BO + OD$ , the diagonals are equal ( $AC = BD$ ).

A quadrilateral with equal diagonals that bisect each other is a rectangle. $\text{ABCD is a rectangle because its diagonals are equal and bisect each other.}$ $\text{All angles of a rectangle are } 90^\circ.$

Step 2 · Properties of the Rectangle

Since $ABCD$ is a rectangle, its opposite sides are equal in length: $AB = CD \quad \text{and} \quad AD = BC$

$\text{As ABCD is a rectangle, its opposite sides must be equal.}$

Step 3 · Effect of Changing the Diagonal Angle

Diagram 1

Let $x$ be the angle between the diagonals at intersection $O$ .

As long as the diagonals remain equal in length and continue to bisect each other, changing the angle $x$ changes the dimensions (side lengths) of $ABCD$ , but the four interior angles remain $90^\circ$ . Hence, $ABCD$ remains a rectangle.
When $x = 90^\circ$ , the diagonals intersect at right angles, making all four sides equal. Thus, the rectangle becomes a square.

$\text{The figure remains a rectangle as long as diagonals are equal and bisect each other.}$ $\text{If the angle between diagonals becomes } 90^\circ, \text{ it becomes a square.}$

Answer

Yes, $ABCD$ remains a rectangle as long as the diagonals are equal and bisect each other. In general, changing the angle $x$ alters the side lengths, and when $x = 90^\circ$ , it becomes a square.

Common Mistakes

Assuming non-rectangles are formed: Believing changing the angle destroys the $90^\circ$ corner angles. As long as diagonals are equal and bisect each other, the figure is guaranteed to be a rectangle.
Square vs. Rectangle: Overlooking that a square is a special type of rectangle where diagonals intersect at $90^\circ$ .

More questions in IT

Observe the following figures.

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

Are there other ways to define a rectangle?

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 —

What is the length of the other diagonal?
What is the point of intersection of the two diagonals?
What should the angle be between the diagonals?

Can the following equalities be used to establish that $\Delta AOD \cong \Delta COB$ ?

$AO = CO$ (proved above)
$\angle AOB = \angle COD$ (vertically opposite angles)
$AD = CB$

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 $60^\circ$ , between them as shown in the figure to the right.

Q. Can you find all the remaining angles?

Context: In $\Delta AOB$ , since $OA = OB$ , the angles opposite them are equal, say $a$ .

Q. Can you find the value of $a$ ?

Can we now identify what type of quadrilateral ABCD is?

Notice that its angles all add up to $90^\circ$ ( $30^\circ + 60^\circ$ ).

What can we say about its sides?

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 $x$ .

Q10

Context: We can compute the four angles between the diagonals to be $x$ , $x$ , $180^\circ - x$ , and $180^\circ - x$ .

Q. Can you find the other angles?

Q11

Context: Since we know that $\triangle AOB$ is isosceles, we can denote the measures of both of its base angles by $a$ .

Q. What is the value of $a$ (in degrees) in terms of $x$ ?

Q12

Context: Thus, all four angles of the quadrilateral $ABCD$ are $90^\circ$ .

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 $90^\circ$ . Would we be wrong if we just define a rectangle as a quadrilateral in which all the angles are $90^\circ$ ?

Q14

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

Are you able to construct such a quadrilateral?

Q15

Is it wrong to write $\Delta BAD \cong \Delta CDB$ ? Why?

Q16

Can you similarly show that $\text{AB}$ is parallel to $\text{DC}$ ( $\text{AB} \parallel \text{DC}$ )?

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 $\angle\text{BOA}$ and $\angle\text{BOC}$ 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 $8\text{ cm}$ .

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 $\angle 2$ and $\angle 4$ .

Q24

4.2 Angles in a Quadrilateral

Is it possible to construct a quadrilateral with three angles equal to $90^\circ$ and the fourth angle not equal to $90^\circ$ ?

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 $ABCD$ with adjacent sides of lengths $4\text{ cm}$ and $5\text{ cm}$ , and an angle of $30^\circ$ 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 $\Delta \text{ABD} \cong \Delta \text{CBD}$ ? 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 $\Delta \text{AOE} \cong \Delta \text{SOY}$ ? 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 $\Delta GAE \cong \Delta MAE$ (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 $\Delta \text{GEO} \cong \Delta \text{MEO}$ (why?).

Q42

In the kite, show that the diagonal $BD$

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

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

Hint: Is $\triangle AOB \cong \triangle COB$ ?

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 $UV \parallel XW$ . Measure $\angle U$ .

Q47

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