Two circles touching internally prove that ox:oy=oa:ob, Mathematics

Two circles touching internally at O. OXY, OAB straight lines, the latter passing through the centres. Prove that OX : OY = OA : OB.

Given : Two circles touching internally

at ‘O' . The line OXY touches the circles
at X and Y . The line OAB passes through
the centres of the circles.
R. T. P.: OX : OY = OA : OB
Construction: Join AX and BY.
Proof : OA is the diametre of the inner
circle.
∠OXA = 900 ( Angle in semicircle )......................( i )
OB is the diameter of the outer circle.

∠OYB = 900 ( Angle in the semi circle...................( ii )

From ( i ) and ( ii ); ∠OXA = ∠OYB =900

i.e., The corresponding angles are equal.
 AX // BY
In ΔOBY; AX //BY
OX/OY = OA/OB

i.e., OX : OY = OA : OB

614_circles.png

Posted Date: 9/3/2012 5:38:00 AM | Location : United States







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