Question

2 right angle triangles ABC and DBC are on the same hypotenuse BC. side AC and BD intersect at P. prove that AP * PC is equal to BP into PD

Answer 1

The quadrilateral formed by ABDC, sum of opposite angles are 180° there it is cyclic quadrilateral.
So from a point P DB and CA are chords.
And using a theorem in circle
PA.PC=PB.PD
Proof for that theorem is in NCERT class 10th.

Answer 2

Answer:

In ∆APB & ∆DPC,

∠A =  ∠D = 90°

∠APB = ∠DPC

[ vertically opposite angles]

∆APB  ~ ∆DPC

[By AA Similarity Criterion ]

AP/DP = PB/ PC

[corresponding sides of similar triangles are proportional]

AP × PC = PB × DP

Hence, proved.