Question

If two equal chords AB and CD of a circle when produced, intersect at point P, then PB =

• A. PD
• B. DY
• C. CY
• D. None of the above

Answer 1

The correct option is A

PD

Given that AB=CD
Draw OX and OY perpendicular to AB and CD respectively

Given , OX = OY (equal chords of a circle are equidistant from the center.)
Conside triangles OXP and OYP,
OX = OY
OP = OP (common)
$\angle OXP=\angle OYP={90}^o$ (they are perpendicular)
$\therefore △OXP\cong OYP$ (by RHS congruency)
PX = PY ( by CPCT).................(i)

We know AX = XB = $\frac{AB}{2}$.....(ii)

CY = YD = $\frac{CD}{2}$....(iii) (the line joining the center of the circle is perpendicular to the chord and bisects the chord.)
From (ii) and (iii)

XB = YD........(iv)

Subtracting (iv) from (i) we get

$\therefore PX-XB=PY-YD$
$\Rightarrow BP=PD$