Classroom activity (Constructing the 'square root spiral') : Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point $ \mathrm{O} $ and draw a line segment $ \mathrm{OP}_{1} $ of unit length. Draw a line segment $ \mathrm{P}_{1} \mathrm{P}_{2} $ perpendicular to $ \mathrm{OP}_{1} $ of unit length (see figure below). Now draw a line segment $ \mathrm{P}_{2} \mathrm{P}_{3} $ perpendicular to $ \mathrm{OP}_{2} $. Then draw a line segment $ \mathrm{P}_{3} \mathrm{P}_{4} $ perpendicular to $ \mathrm{OP}_{3} $. Continuing in Fig. 1.9: Constructing this manner, you can get the line segment $ \mathrm{P}_{\mathrm{a}-1} \mathrm{P}_{\mathrm{n}} $ by square root spiral drawing a line segment of unit length perpendicular to $ \mathrm{OP}_{\mathrm{n}-1} $. In this manner, you will have created the points $ \mathrm{P}_{2}, \mathrm{P}_{3}, \ldots, \mathrm{P}_{\mathrm{n}}, \ldots . $, , and joined them to create a beautiful spiral depicting $ \sqrt{2}, \sqrt{3}, \sqrt{4}, \ldots $
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To do:
We have to construct a square root spiral as given in the question.
Solution:
Steps of construction:
1. Mark a point $A$ on a paper.
$A$ is the center of the square root spiral.
2. From $A$, draw a straight line $AB$ of $1\ cm$ horizontally.
3. From $B$, draw a perpendicular line $AC$ of $1\ cm$.
4. Join $AC$.
$AC^2=AB^2+BC^2$
$AC^2=1^2+1^2$
$AC=\sqrt{2}\ cm$
5. From $C$, draw a perpendicular line of $1\ cm$ and mark the end point $D$.
6. Join $AD$. $AD=\sqrt{3}\ cm$
7. Similarly, $AE=\sqrt{4}\ cm, AF=\sqrt5\ cm,.......$
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