[Śulba-Sūtras] An isosceles triangle can be converted into a rectangle by dissection in a simpler way. Can you find out how to do it? [Hint: Show that triangles and can be made into halves of a rectangle. Figure out how they should be assembled to get a rectangle. Use cut-outs if necessary.]

Understand the Question An isosceles triangle has two equal sides and a base . Drawing the altitude divides the triangle into two congruent right-angled triangles: and . A rectangle of length equal to altitude and width equal to half the base (or ) has the exact same area as the triangle: . By cutting along the altitude and joining the two right-angled triangles along their hypotenuses ( and ), we can assemble them into a rectangle. Step 1 · Dissect the Isosceles Triangle Consider an isosceles triangle where , and draw the altitude . In and : (Given) (Common side) By RHS congruence: Therefore: Cutting along divides into two congruent right-angled triangles. Step 2 · Rearrange the Halves to Form a Rectangle Let altitude and half-base . The area of the original triangle is: To assemble the rectangle: 1. Keep fixed with base and height . 2. Rotate by and align hypotenuse along hypotenuse . 3. The two right triangles combine to form a rectangle of dimensions (or ). Answer > Cut the isosceles triangle along its altitude into two congruent right-angled triangles and , then rearrange them by joining their equal hypotenuses ( and ) to form a rectangle of dimensions . Common Mistakes Joining Along the Legs: Merely placing the two halves side by side along leg or base reconstructs the original triangle or another non-rectangular shape instead of a rectangle. Hypotenuse Alignment: Forgetting that one triangle must be rotated/flipped so that the two equal hypotenuses match, forming opposite parallel sides with right angles at all four corners.

Area of Polygons | FIO

Question 26

[Śulba-Sūtras] An isosceles triangle can be converted into a rectangle by dissection in a simpler way. Can you find out how to do it?

[Hint: Show that triangles $\Delta \text{ADB}$ and $\Delta \text{ADC}$ can be made into halves of a rectangle. Figure out how they should be assembled to get a rectangle. Use cut-outs if necessary.]

Solution

Understand the Question

An isosceles triangle $\Delta \text{ABC}$ has two equal sides $\text{AB} = \text{AC}$ and a base $\text{BC}$ .
Drawing the altitude $\text{AD} \perp \text{BC}$ divides the triangle into two congruent right-angled triangles: $\Delta \text{ADB}$ and $\Delta \text{ADC}$ .
A rectangle of length equal to altitude $\text{AD}$ and width equal to half the base $\text{BD}$ (or $\text{DC}$ ) has the exact same area as the triangle: $\text{Area} = b \times h$ .
By cutting along the altitude $\text{AD}$ and joining the two right-angled triangles along their hypotenuses ( $ext{AB}$ and $ext{AC}$ ), we can assemble them into a rectangle.

Step 1 · Dissect the Isosceles Triangle

Consider an isosceles triangle $\Delta \text{ABC}$ where $\text{AB} = \text{AC}$ , and draw the altitude $\text{AD} \perp \text{BC}$ . Diagram 1

In $\Delta \text{ADB}$ and $\Delta \text{ADC}$ :

$\angle \text{ADB} = \angle \text{ADC} = 90^\circ$
$\text{AB} = \text{AC}$ (Given)
$\text{AD} = \text{AD}$ (Common side)

By RHS congruence: $\Delta \text{ADB} \cong \Delta \text{ADC}$

Therefore: $\text{BD} = \text{DC}$

Cutting along $\text{AD}$ divides $\Delta \text{ABC}$ into two congruent right-angled triangles.

Step 2 · Rearrange the Halves to Form a Rectangle

Let altitude $\text{AD} = h$ and half-base $\text{BD} = \text{DC} = b$ .

The area of the original triangle is:

\begin{aligned} \text{Area}(\Delta \text{ABC}) &= \dfrac{1}{2} \times \text{base} \times \text{height} \\[0.6em] &= \dfrac{1}{2} \times \text{BC} \times \text{AD} \\[0.6em] &= \dfrac{1}{2} \times (\text{BD} + \text{DC}) \times \text{AD} \\[0.6em] &= \dfrac{1}{2} \times (b + b) \times h \\[0.6em] &= \dfrac{1}{2} \times 2b \times h \\[0.6em] &= b \times h \end{aligned}

To assemble the rectangle:

Keep $\Delta \text{ADC}$ fixed with base $\text{DC} = b$ and height $\text{AD} = h$ .
Rotate $\Delta \text{ADB}$ by $180^\circ$ and align hypotenuse $\text{AB}$ along hypotenuse $\text{AC}$ .
The two right triangles combine to form a rectangle of dimensions $b \times h$ (or $\text{DC} \times \text{AD}$ ).

Answer

Cut the isosceles triangle along its altitude $\text{AD}$ into two congruent right-angled triangles $\Delta \text{ADB}$ and $\Delta \text{ADC}$ , then rearrange them by joining their equal hypotenuses ( $\text{AB}$ and $\text{AC}$ ) to form a rectangle of dimensions $\text{AD} \times \text{DC}$ .

Common Mistakes

Joining Along the Legs: Merely placing the two halves side by side along leg $\text{AD}$ or base $\text{BD}$ reconstructs the original triangle or another non-rectangular shape instead of a rectangle.
Hypotenuse Alignment: Forgetting that one triangle must be rotated/flipped so that the two equal hypotenuses match, forming opposite parallel sides with right angles at all four corners.

More questions in FIO

Identify the missing sidelengths.

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?

The figure shows a plot with sides $14\text{ m}$ and $12\text{ m}$ , and with a crosspath. What other measurements do you need to find the area of the crosspath? Once you identify the lengths to be measured, assign some possible values of your choice and find the area of the path. Give a formula for the area based on the measurements you choose.

Find the area of the spiral tube shown in the figure. The tube has the same width throughout.

[Hint: There are different ways of finding the area. Here is one method.]

What should be the length of the straight tube if it is to have the same area as the bent tube on the left?

In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons.

Divide a square into 4 parts by drawing two perpendicular lines inside the square as shown in the figure.

Rearrange the pieces to get a larger square, with a hole inside.

You can try this activity by constructing the square using cardboard, thick chart paper, or similar materials.

Find the areas of the following triangles:

Find the length of the altitude $\text{BY}$ .

Find the area of $\Delta\text{SUB}$ , given that it is isosceles, $\text{SE}$ is perpendicular to $\text{UB}$ , and the area of $\Delta\text{SEB}$ is $24\text{ sq. units}$ .

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

If $M$ and $N$ are the midpoints of $XY$ and $XZ$ , what fraction of the area of $\Delta\text{XYZ}$ is the area of $\Delta\text{XMN}$ ? [Hint: Join $NY$ ]

Q14

Gopal needs to carry water from the river to his water tank. He starts from his house. What is the shortest path he can take from his house to the river and then to the water tank? Roughly recreate the map in your notebook and trace the shortest path.

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

What measurements would you need to find the area of a regular hexagon?

Q18

What fraction of the total area of the rectangle is the area of the blue region?

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

Find $\text{QN}$ .

Q23

Consider a rectangle and a parallelogram of the same sidelengths: $5\text{ cm}$ and $4\text{ cm}$ . Which has the greater area? [Hint: Imagine constructing them on the same base.]

Q24

Give a method to obtain a rectangle whose area is twice that of a given triangle. What are the different methods that you can think of?

Q25

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

Q26

[Śulba-Sūtras] An isosceles triangle can be converted into a rectangle by dissection in a simpler way. Can you find out how to do it?

[Hint: Show that triangles $\Delta \text{ADB}$ and $\Delta \text{ADC}$ can be made into halves of a rectangle. Figure out how they should be assembled to get a rectangle. Use cut-outs if necessary.]

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

Give a method to convert a rectangle into a rhombus of equal area using dissection.

Q31

Find the areas of the following figures:

Q32

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

Q33

Here is one of the ways to convert trapezium $ABCD$ into a rectangle $EFGH$ of equal area —

Given the trapezium $ABCD$ , how do we find the vertices of the rectangle $EFGH$ ?

[Hint: If $\Delta AHI \cong \Delta DGI$ and $\Delta BEJ \cong \Delta CFJ$ , then the trapezium and rectangle have equal areas.]

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

A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas.

Q36

ZYXW is a trapezium with $ZY \parallel WX$ . $A$ is the midpoint of $XY$ . Show that the area of the trapezium ZYXW is equal to the area of $\triangle ZWB$ .