Q1.
Try working this out! [Transform a rhombus into a rectangle of the same area by dissection, by the method that occurs in one of the Śulba-Sūtras.]
Answer
A rhombus is a parallelogram, so base × height already works. But its extra properties give a neater dissection.
- In rhombus ABCD, draw both diagonals. They meet at O, and — this is the key — they are perpendicular bisectors of each other.
- The diagonal BD cuts the rhombus into ∆ABD and ∆CBD. Since AB = AD and CB = CD, both are isosceles triangles on the base BD.
- Convert each isosceles triangle into a rectangle by the dissection of Q7 (page 164): cut along its axis of symmetry (AO for the first, CO for the second) and fit the two right-triangle halves together along their slant sides.
- Join the two rectangles side by side. They fit, because both have the same height, ½BD.
Result: a single rectangle WXYZ with
XW = AO + OC = AC and WZ = ½ BD
Area = AC × ½BD = ½ × AC × BD
XW = AO + OC = AC and WZ = ½ BD
Area = AC × ½BD = ½ × AC × BD
Why it happens: the two diagonals of a rhombus meet at right angles, which is exactly what makes ∆ABD and ∆CBD isosceles with BD as base — and an isosceles triangle is the easiest shape of all to turn into a rectangle. Every step is a cut and a move, so the area never changes.