Find the area of the quadrilateral given that , , , is perpendicular to , and is perpendicular to .

Understand the Question A quadrilateral can be divided into two triangles along its diagonal. The diagonal divides quadrilateral into two triangles: and . Given that and , and serve as the perpendicular heights for and on the common base . Total area of quadrilateral . Step 1 · Calculate Area of Triangle ABC For , base and height . Step 2 · Calculate Area of Triangle ADC For , base and height . Step 3 · Calculate Total Area of Quadrilateral ABCD Add the areas of and Answer > Common Mistakes Forgetting the Factor: Using instead of when calculating the area of each triangle. Incorrect Formula: Applying general quadrilateral formulas without verifying that is the common base for both perpendicular offsets and . Missing Units: Writing the final answer without area units ().

Area of Polygons | FIO

Question 15

Find the area of the quadrilateral $ABCD$ given that $AC = 22\text{ cm}$ , $BM = 3\text{ cm}$ , $DN = 3\text{ cm}$ , $BM$ is perpendicular to $AC$ , and $DN$ is perpendicular to $AC$ .

Solution

Understand the Question

A quadrilateral can be divided into two triangles along its diagonal.
The diagonal $AC$ divides quadrilateral $ABCD$ into two triangles: $\Delta ABC$ and $\Delta ADC$ .
Given that $BM \perp AC$ and $DN \perp AC$ , $BM$ and $DN$ serve as the perpendicular heights for $\Delta ABC$ and $\Delta ADC$ on the common base $AC$ .
$\text{Area of a triangle} = \dfrac{1}{2} \times \text{base} \times \text{height}$
Total area of quadrilateral $ABCD = \text{Area}(\Delta ABC) + \text{Area}(\Delta ADC)$ .

Step 1 · Calculate Area of Triangle ABC

Diagram 1

For $\Delta ABC$ , base $= AC = 22\text{ cm}$ and height $= BM = 3\text{ cm}$ .

\begin{aligned} \text{Area of triangle ABC} &= \dfrac{1}{2} \times AC \times BM \\[0.6em] &= \dfrac{1}{2} \times 22\text{ cm} \times 3\text{ cm} \\[0.6em] &= 11\text{ cm} \times 3\text{ cm} \\[0.6em] &= 33\text{ cm}^2 \end{aligned}

Step 2 · Calculate Area of Triangle ADC

For $\Delta ADC$ , base $= AC = 22\text{ cm}$ and height $= DN = 3\text{ cm}$ .

\begin{aligned} \text{Area of triangle ADC} &= \dfrac{1}{2} \times AC \times DN \\[0.6em] &= \dfrac{1}{2} \times 22\text{ cm} \times 3\text{ cm} \\[0.6em] &= 11\text{ cm} \times 3\text{ cm} \\[0.6em] &= 33\text{ cm}^2 \end{aligned}

Step 3 · Calculate Total Area of Quadrilateral ABCD

Add the areas of $\Delta ABC$ and $\Delta ADC$

\begin{aligned} \text{Area of quadrilateral ABCD} &= \text{Area of triangle ABC} + \text{Area of triangle ADC} \\[0.6em] &= 33\text{ cm}^2 + 33\text{ cm}^2 \\[0.6em] &= 66\text{ cm}^2 \end{aligned}

Answer

$66\text{ cm}^2$

Common Mistakes

Forgetting the $\dfrac{1}{2}$ Factor: Using $\text{base} \times \text{height}$ instead of $\dfrac{1}{2} \times \text{base} \times \text{height}$ when calculating the area of each triangle.
Incorrect Formula: Applying general quadrilateral formulas without verifying that $AC$ is the common base for both perpendicular offsets $BM$ and $DN$ .
Missing Units: Writing the final answer without area units ( $\text{cm}^2$ ).

More questions in FIO

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The figure shows a path (the shaded portion) laid around a rectangular park EFGH.

(i) What measurements do you need to find the area of the path? Once you identify the lengths to be measured, assign possible values of your choice to these measurements and find the area of the path. Give a formula for the area.

An example of a formula — $\text{Area of a rectangle} = \text{length} \times \text{width}$ .

[Hint: There is a relation between the areas of EFGH, the path, and ABCD.**]

(ii) If the width of the path along each side is given, can you find its area? If not, what other measurements do you need? Assign values of your choice to these measurements and find the area of the path. Give a formula for the area using these measurements.

[Hint: Break the path into rectangles.**]

(iii) Does the area of the path change when the outer rectangle is moved while keeping the inner rectangular park EFGH inside it, as shown?

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Q10

[Śulba-Sūtras] Give a method to transform a rectangle into a triangle of equal area.

Q11

[Śulba-Sūtras] Give a method to transform a triangle into a rectangle of equal area.

Q12

ABCD, BCEF, and BFGH are identical squares.

(i) If the area of the red region is 49 sq. units, then what is the area of the blue region?

(ii) In another version of this figure, if the total area enclosed by the blue and red regions is 180 sq. units, then what is the area of each square?

Q13

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Q14

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Q15

Find the area of the quadrilateral $ABCD$ given that $AC = 22\text{ cm}$ , $BM = 3\text{ cm}$ , $DN = 3\text{ cm}$ , $BM$ is perpendicular to $AC$ , and $DN$ is perpendicular to $AC$ .

Q16

Find the area of the shaded region given that ABCD is a rectangle.

Q17

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Q18

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Q19

Give a method to obtain a quadrilateral whose area is half that of a given quadrilateral.

Q20

Observe the parallelograms in the figure below.

(i) What can we say about the areas of all these parallelograms?

(ii) What can we say about their perimeters? Which figure appears to have the maximum perimeter, and which has the minimum perimeter?

Q21

Find the areas of the following parallelograms:

Q22

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Q23

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Q24

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Q25

[Śulba-Sūtras] Give a method to obtain a rectangle of the same area as a given triangle.

Q26

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Q27

[Śulba-Sūtras] Give a method to convert a rectangle into an isosceles triangle by dissection.

Q28

Which has greater area—an equilateral triangle or a square of the same sidelength as the triangle? Which has greater area—two identical equilateral triangles together or a square of the same sidelength as the triangle? Give reasons.

Q29

Find the area of a rhombus whose diagonals are $20\text{ cm}$ and $15\text{ cm}$ .

Q30

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Q31

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Q32

[Śulba-Sūtras] Give a method to convert an isosceles trapezium to a rectangle using dissection.

Q33

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Q34

Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of area $144\text{ cm}^2$ .

Q35

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Q36

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