Constructing a rectangle (one given side) of area equal to the area of a given square
- Author:
- Amaresh G S
- Topic:
- Area, Constructions, Geometric Mean, Geometry, Rectangle, Similar Triangles, Square
Method based on a Āpastamba Śulbasūtra.
समचतुरश्रं दीर्घचतुरश्रं चिकीर्षन् यावच्चिकीर्षेत् तावर्तीं पर्श्वमानीं कृत्वा यदधिकं स्यात् यथायोगमुपदध्यात् । (Āpastamba Śulbasūtra III. 1)
Sundararāja, the commentator of Āpastamba Śulbasūtra, explaining this sūtra, gives a purely geometrical and exact construction.
यावदिच्छं पर्श्वमान्यौ प्राच्यौ वर्धयित्वा उत्तरपूर्वां कर्णरज्जुमायच्छेत्, सा दीर्घचतुरश्रमध्यस्थायां समचतुरश्रतिर्यङ्मान्यां यत्र निपतति तत उत्तरं हित्वा दक्षिणांशां तिर्यङ्मानीं कुर्यात्, तद् दीर्घचतुरश्रं भवति ।
Ref: Geometry in ancient and medieval India. - T.A. Sarasvati Amma.
The area of the square can be changed by dragging the point 'B'.
The measure of a side of the rectangle can be changed by dragging the point 'I'.
Another method.
This method is based on the below fact.
In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of these two segments.
The area of the square can be changed by dragging the point 'B'.
The measure of a side of the rectangle can be changed by dragging the point 'E'.