Quadrilaterals | IT

Question 40

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

Solution

The diagonals of a rhombus always meet at a right angle.

Step 1 — Properties of a Rhombus

Let us consider a rhombus ABCD. A rhombus is a quadrilateral. All its four sides are equal in length. So, $AB = BC = CD = DA$ . A rhombus is also a type of parallelogram. Its diagonals bisect each other. Let diagonals AC and BD intersect at point O. So, $AO = OC$ and $BO = OD$ .

Diagram 1

Step 2 — Using Congruent Triangles

Let us look at two triangles: $\triangle AOB$ and $\triangle COB$ . We know $AB = BC$ . (These are sides of the rhombus). We know $AO = OC$ . (The diagonals bisect each other). The side BO is common to both triangles. So, $BO = BO$ . These three sides are equal. This means $\triangle AOB$ is congruent to $\triangle COB$ . (Congruent means identical in shape and size). We use the SSS (Side-Side-Side) congruence rule. Since the triangles are congruent, their corresponding angles are equal. So, $\angle AOB = \angle COB$ .

Step 3 — Finding the Angle

Angles $\angle AOB$ and $\angle COB$ are adjacent angles. They form a linear pair on the straight line AC. A linear pair means they add up to 180 degrees. So, we can write the equation: $\angle AOB + \angle COB = 180^\circ$ From Step 2, we know $\angle AOB = \angle COB$ . Let us substitute $\angle AOB$ for $\angle COB$ : $\angle AOB + \angle AOB = 180^\circ$ $2 \times \angle AOB = 180^\circ$ $\angle AOB = \frac{180^\circ}{2}$

$\boxed{\angle AOB = 90^\circ}$ This means the diagonals intersect at a right angle.

Answer

The diagonals of a rhombus intersect at a right angle (90 degrees).

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° (30° + 60°).

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 $x, x, 180 - x,$ and $180 - x.$

Q. Can you find the other angles?

Q11

Context: Since we know that $\Delta AOB$ 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°.

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 $\angle\text{BOA}$ 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 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° 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 $ABCD$ 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?