Consider the given figure.
If \(CE\) is perpendicular to \(PQ\), \(CH\) is perpendicular to \(RS\), and \(CE = CH\), confirm that \(PQ = RS\) using the Baudhāyana–Pythagoras theorem.
Proof:
Since, \(CE\) is perpendicular to \(PQ\), the triangles \(CEQ\) and \(CEP\) are right angled triangles.
Applying Pythagoras theorem, we get:
In \(\Delta CEQ\), \(= CE^2 + EQ^2\) ......(1)
In \(\Delta CEP\), \(= CE^2 + EP^2\) ......(2)
Adding (1) and (2), we get:
\(= 2CQ^2 - 2CE^2\) ......(3)
Similarly, since, \(CH\) is perpendicular to \(RS\), the triangles \(CHR\) and \(CHS\) are right angled triangles.
Applying Pythagoras theorem, we get:
In \(\Delta CHR\), \(= CH^2 + HR^2\) ......(4)
In \(\Delta CHS\), \(= CH^2 + HS^2\) ......(5)
\(= 2CQ^2 - 2CE^2\) ......(6)
Equating (3) and (6), we get:
\(=\)
Taking square root on both sides, we get:
\(PQ = RS\)
Hence, proved.