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Question

As shown in the fig. 3.14 a semi circle with centre O and diameter AB is drawn. Keeping the radius constant, arcs are drawn with points A and B as centres. These arcs intersect the semi circle at points P and Q respectively. Similarly two more arcs are drawn. keeping radius constant, with centres P and Q. These arcs intersect at point X. Show that seg XO line AB.

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Solution



Given : AB is the diameter.All arcs are drawn with the same radius.Construction : Join PX, QX, PA, QB, PO and QO.In AOP and BOQ,AO = OB (O is the midpoint)AP = QB (arcs are cut with same radius)PO =QO radius of the same circle AOP BOQ (SSS test)Thus, AOP = BOQ (by c.a.c.t) ....(i)In POX and QOX,PO = QO radius of the same circlePX = QX (arcs are cut with same radius)OX = OX (common)POX QOX (SSS test)Thus, POX = QOX (by c.a.c.t) ......(ii)Adding (i) and (ii), we get:AOX = BOX

Since , AOX + BOX = 180° linear pair2AOX =180°AOX = 90° = BOX

Hence, seg XO is perpendicular to line AB.

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