Question

If PQ and RS are two chords of a circle with center O intersecting each other. If X be any point on RS such that XO//SQ prove that PX =SX

Answer 1

Answer:

The most critical part of this question is to get the diagram right. Once the figure is correctly drawn, the solution becomes extremely simple.

Please, refer to the figure attached while referring to the solution below:

O is the centre of the circle and XO // SQ ( Given information)

Let us join S and O.

OS = OQ (both are radius of the circle)

∠OQS =  ∠OSQ ( because OS = OQ )   ...........statement 1

∠OQS = ∠POS (corresponding angles as XO // SQ) ......... statement 2

∠XOS = ∠OSQ ( Alternate angles as XO // SQ)............... statement 3

∴ From above 3 statements, we can conclude that

∠XOS = ∠POS   .............statement 4

IN ΔPXO & ΔOXS,

PO = OS (radius of the circle)

∠POS = ∠XOS   (from statement 4)

Side OX = Side OX (common side of both traingles)

∴ By SAS postulate ( side angle side) we can say that,

ΔPXO ≅ ΔOXS

Now, we know that corresponding sides of congruent sides are equal,

∴ PX = SX

Hence Proved.