Area of Polygons | IT

Question 18

Try working this out!

Solution

Understand the Question

A quadrilateral with perpendicular diagonals intersecting at $O$ is divided into four right-angled triangles.
Given that diagonal $AC$ bisects diagonal $BD$ ( $BO = OD$ ), each pair of adjacent triangles sharing a segment of $AC$ can be proven congruent.
We use the SAS (Side-Angle-Side) congruence criterion: two pairs of equal sides and the included right angle ( $90^\circ$ ).

Step 1 · Identify Given Properties from the Figure

In quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at point $O$ . Diagram 1

From the markings on the figure:

Diagonals are perpendicular: $\angle AOB = \angle AOD = \angle BOC = \angle DOC = 90^\circ$
Bisected diagonal: $BO = OD$
Equal side lengths: $AB = AD$ and $BC = CD$

Step 2 · Prove Congruence of Triangles

For $\triangle AOB$ and $\triangle AOD$ :

\begin{aligned} AO &= AO \quad (\text{Common side}) \\ \angle AOB &= \angle AOD = 90^\circ \\ BO &= OD \quad (\text{Given}) \end{aligned}

By SAS congruence criterion: $\triangle AOB \cong \triangle AOD$ Diagram 2

For $\triangle BOC$ and $\triangle DOC$ :

\begin{aligned} CO &= CO \quad (\text{Common side}) \\ \angle BOC &= \angle DOC = 90^\circ \\ BO &= OD \quad (\text{Given}) \end{aligned}

By SAS congruence criterion: $\triangle BOC \cong \triangle DOC$

Answer

$\text{(i) } \triangle AOB \cong \triangle AOD, \quad \text{(ii) } \triangle BOC \cong \triangle DOC$

Common Mistakes

SAS vs. RHS Criterion: Mistaking the proof for RHS simply because right angles are involved; since the included angle lies between the two known sides ( $AO/CO$ and $BO/OD$ ), the correct criterion is SAS.
Vertex Order: Writing incorrect corresponding vertices in the congruence statement, such as $\triangle AOB \cong \triangle ADO$ instead of $\triangle AOB \cong \triangle AOD$ .

More questions in IT

Try to think of different creative ways to divide a square into 4 parts of equal area.

Why Can't Perimeter be a Measure of Area?

Why do we count the number of unit squares to assign measures for area? Couldn't we have just used the perimeter of a region, i.e., the length of its boundary as a measure of its area?

Context: Consider two regions, Region 1 and Region 2, such that $\text{Perimeter of Region 1} > \text{Perimeter of Region 2}$ , but $\text{Area of Region 1} < \text{Area of Region 2}$ .

Q. Find two rectangles that are examples of such regions. If needed, use a grid paper (given at the end of the book) for this.

Also give an example of two regions of other shapes, where the region with the larger perimeter has the smaller area! This property should be visually clear in your example.

In the given figure, which triangle has a greater area: $\Delta \text{XDC}$ or $\Delta \text{YDC}$ , if both the rectangles are identical?

In the given figure, which triangle has a greater area: $\Delta \text{XDC}$ or $\Delta \text{YBC}$ , if both the rectangles are identical?

Find the area of $\Delta \text{XDC}$ .

To find the area of a triangle, what measurements do we need?

How do we get the outer rectangle from the given triangle?

Q10

Will this formula hold for the kind of triangle, around which we cannot draw a rectangle with $\text{BC}$ as the base?

Q11

Line $l \parallel \text{BC}$ . Consider the different triangles that have $\text{BC}$ as their base, and with their third vertex lying anywhere on $l$ .

(i) Which of these triangles has the maximum area, and which has the minimum area?

(ii) Which of these triangles has the maximum perimeter, and which has the minimum perimeter?

Q12

Analyse whether $A$ lies on the perpendicular bisector of $BC$ .

Q13

Area of any Polygon

How do we find the area of this quadrilateral? What measurements do we need for this?

Q14

How do we find the area of this pentagon?

Q15

Can any polygon be divided into triangles?

Q16

Give a method to convert a parallelogram into a rectangle of equal area.

You can try this using a cut-out of a parallelogram.

Q17

Can $\Delta \text{AXD}$ and $\text{ABCX}$ fit together, as shown in the figure, to get a rectangle?

Q18

Try working this out!

Q19

What are the sidelengths of the rectangle $WXYZ$ ?

Q20

Area of rhombus ABCD can also be determined by finding the areas of $\Delta \text{ADB}$ and $\Delta \text{CDB}$ . What formula does this give us?

Q21

Context: The area of rhombus ABCD can be written as:

$\text{Area of rhombus } ABCD = \text{Area}(\Delta \text{ADB}) + \text{Area}(\Delta \text{CDB}) = \dfrac{1}{2} \times \text{AO} \times \text{BD} + \dfrac{1}{2} \times \text{CO} \times \text{BD}$

Q. Simplify the expression to show that we get the same formula for the area of a rhombus in terms of its diagonals.

Q22

Find the areas of the following trapeziums by breaking them into figures whose areas can be computed.

Q23

Will this formula hold for a trapezium that looks like this?

Q24

Will Approach 2 work for any type of trapezium?

Q25

What figure will we get when the two trapeziums are joined along $\text{BC}$ ?

Q26

What type of a quadrilateral is this?

Q27

What do you think is the area of an A4 sheet? Its sidelengths are $21\text{ cm}$ and $29.7\text{ cm}$ . Now find its area.

Q28

What do you think is the area of the tabletop that you use at school or at home? You could perhaps try to visualise how many A4 sheets can fit on your table.

Q29

Express the following lengths in centimeters:

(i) $5 \text{ in}$

(ii) $7.4 \text{ in}$

Q30

Express the following lengths in inches:

(i) $5.08 \text{ cm}$

(ii) $11.43 \text{ cm}$

Q31

How many $\text{cm}^2$ is $1 \text{ in}^2$ ?

Q32

Context: Convert $161.29 \text{ cm}^2$ to $\text{in}^2$ . Every $6.4516 \text{ cm}^2$ gives an $\text{in}^2$ . Hence, $161.29 \text{ cm}^2 = \dfrac{161.29}{6.4516} \text{ in}^2$ .