CAT Some Results on Chord and Circle - Practice Questions & MCQ

Result 1: If two chords of a circle intersect each other, then the products of the lengths of their segments are equal.

Let us prove this

Case 1: Let the two chords intersect internally (inside the circle).

Let AB and CD are two chords intersecting at point P in the circle, and we have to prove (PA) (PB) = (PC) (PD).

Join AC and BD.

In triangles APC and PDB,

∠APC = ∠DPB                   (Vertically opposite angels)

∠CAP = ∠CDB                   (Angles made by arc BC in the same segment)

ΔAPC is similar to ΔDPB.    (by the AA Similarity Property)

Case 2: Let the two chords intersect externally (outside the circle).

Let two chords BA and CD intersect at point P which lies outside the circle, and we have to prove (PA) (PB) = (PC) (PD).

Join AC and BD.  

In triangles PAC and PDB,

       ∠PAC = ∠PDB

and ∠PCA = ∠PBD

(An external angle of a cyclic quadrilateral is equal to the interior angle at the opposite vertex)

ΔPAC is similar to ΔPDB.    (by the AA Similarity Property)

Result 2: The converse of the above Result is also true, i.e., if two line segments AB and CD intersect at P and (PA)(PB) = (PC)(PD), then the four points are concyclic.

Result 3: If one of the secants (say PCD) is rotated around P so that it becomes a tangent, i.e., points C and D say at T, coincide. We get the following result:

If PAB is a secant to a circle intersecting the circle at A and B and PT is the tangent drawn from P to the circle, then PA · PB = PT²