Q22 of 141 Page 464

In the give figure, OPQR is a square. A circle drawn with centre O cuts the square in x and y. Prove that QX = XY.

Construction: Join OX and OY

In Δ OPX and ΔORY,

OX = OY (radii of the same circle)

OP = OR (sides of the square)

∴ΔOPX Δ ORY (RHS rule)

∴ PX = RY (CPCT)—1

OPQR is a square

∴ PQ = RQ

∴ PX + QX = RY + QY

QX = QY (from –1)

Hence proved

In the given figure, O is the centre of a circle and ∠BCO = 30°. Find x and y.

PQ and RQ are the chords of a circle equidistant from the centre. Prove that the diameter passing through Q bisects ∠PQR and ∠PSR.

Prove that there is one and only one circle passing through three non – collinear points.

In the given figure, AB and AC we two equal chords of a circle with centre O. Show that O lies on the bisectors of ∠BAC.