In Fig. 6.61, two chords AB and CD intersect each other at the point P. Prove that:
(i) ΔAPC ~ ΔDPB (ii) AP.PB = CP.DP
Solution:
As we know that, two triangles, are similar if

Their corresponding angles are equal and

Their corresponding sides are in the same ratio
As we know that angles in the same segment of a circle are equal.
(i) Draw BC
In ΔAPC and ΔDPB
∠APC = ∠DPB (Vertically opposite angles)
∠PAC = ∠PDB (Angles in the same segment)
⇒ ΔAPC ~ ΔDPB (A.A criterion)
(ii) In ΔAPC and ΔDPB,
AP/DP = CP/PB = AC/DB [∵ ΔAPC ~ ΔDPB]
AP/DP = CP/PB
⇒ AP.PB = CP.DP
Video Solution:
In Fig. 6.61, two chords AB and CD intersect each other at the point P. Prove that: (i) ΔAPC ~ ΔDPB (ii) AP.PB = CP.DP
NCERT Class 10 Maths Solutions  Chapter 6 Exercise 6.6 Question 7:
In Fig. 6.61, two chords AB and CD intersect each other at the point P. Prove that: (i) ΔAPC ~ ΔDPB (ii) AP.PB = CP.DP
In the above figure, two chords AB and CD intersect each other at point P. Hence it is proved that ΔAPC ~ ΔDPB and AP.PB = CP.DP