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Ratio of lengths in right-angled triangle

Source: South African Mathematics Olympiad 2021, Problem 2

August 11, 2021

ratiogeometryright triangle

Problem Statement

Let

PAB

and

PBC

be two similar right-angled triangles (in the same plane) with

\angle PAB = \angle PBC = 90^\circ

such that

A

and

C

lie on opposite sides of the line

PB

. If

PC = AC

, calculate the ratio

\frac{PA}{AB}