Question

Line AB and line XY are diameters of circle O. Prove that AX = BY​

Answer 1

The two lines AB and XY are of equal length.

• A circle is defined as the geometrical figure which is formed by joining the points which are equidistant from a fixed point.
• A circle consists of properties like radius, diameter, circumference, arc length, etc.
• The radius of a circle is defined as the length between the center of the circle and the boundary of the circle. The radius is always the same irrespective of the point on the circle to the center.
• The diameter of a circle is defined as twice the radius of the circle, diameter is obtained by joining any two points which form a straight line passing through the center of the circle.
• The diameter of a circle is always fixed, Hence the line can move through any point on the circle but should necessarily touch/move through the center of the circle.

Therefore, The lines AB and XY which are diameters to a circle have equal lengths

Answer 2

Proved that AX = BY if Line AB and line XY are diameters of circle

Diameters of a circle passes through center.

Assume O is the center

Then AB and XY intersect at O

AB = XY  ( Diameters of circle)

∠AOX = ∠BOY  ( vertically opposite angles)

An inscribed angle is half of a central angle that subtends the same arc

∠ABX = (1/2)∠AOX  

∠YXB = (1/2) ∠BOY  

Hence ∠ABX = ∠YXB

in ΔABX  and  ΔYXB

AB = YX       Diameter

∠ABX = ∠YXB  (already shown)

BX  = XB       Common

=> ΔABX  ≅  ΔYXB  ( using SAS congruence)

Corresponding parts of congruent triangles are Equal in measures

Hence AX =  BY

QED

Hence Proved

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