A square is inscribed in a right-angled triangle with legs p and q, and has a common right angle with the triangle. The diagonal of the square is given by

In ΔAPQ and Δ ABC,

∠APQ = ∠ABC = 90°

∠A is common.

By AA-similarity,

ΔAPQ ≅ ΔABC

Let, PB = PQ = QR = BR

= x units

$frac{mathrm{PQ}}{mathrm{BC}}=frac{mathrm{AP}}{mathrm{AB}}$

$Rightarrow frac{x}{q}=frac{p-x}{p}$

$Rightarrow p x=p q-q x$

$Rightarrow p x+q x=p q$

$Rightarrow x(p+q)=p q$

$Rightarrow x=frac{p q}{p+q}$

$herefore mathrm{BQ}=sqrt{2} x=frac{sqrt{2} p q}{p+q}$ units