Problem
GEO-B2-M04-P024 Choosing the Correct Correspondence
For a point \(P\), it is known that \(\angle APC=\angle BPD\), \(PA=6\), \(PB=9\), \(PC=10\), \(PD=15\). Prove that \(P\) is the centre of a spiral similarity sending \(AC\) to \(BD\), and find the ratio \(AC:BD\).
Check the proportion \(\frac{PA}{PB}=\frac{PC}{PD}\), then apply similarity.
We have \(\frac{PA}{PB}=\frac{6}{9}=\frac{2}{3}\) and \(\frac{PC}{PD}=\frac{10}{15}=\frac{2}{3}\). By the condition, \(\angle APC=\angle BPD\). Hence \(\triangle PAC \sim \triangle PBD\) by two proportional sides and the included angle. Therefore \(P\) is the centre of the spiral similarity sending \(AC\) to \(BD\). From similarity, \(\frac{AC}{BD}=\frac{PA}{PB}=\frac{2}{3}\), so \(AC:BD=2:3\).
The final problem checks whether the student can choose the correct pairs of sides, not merely notice equal numbers.