Let A, B be the centres of two circles of equal radii; draw them so that each one of them passes through the centre of the other. Let them intersect at $ C $ and $ D $. Examine whether $ \overline{\mathrm{AB}} $ and $ \overline{\mathrm{CD}} $ are at right angles.
Given:
$A, B$ are the centres of two circles of equal radii.
To do:
We have to examine whether $\overline{AB}$ and $\overline{CD}$ are at right angles.
Solution:
Let us draw two circles of equal radii so that each one of them passes through the center of the other.
Mark the centres of these circles as $A$ and $B$ respectively.
The circles intersect each other at $C$ and $D$ respectively.
Join $AC, BC, BD, AD$
Therefore,
A quadrilateral $ADBC$ is formed.
From the quadrilateral $ADBC$,
$AD = AC = BC = BD$ (Radius of both the circles is equal)
Hence, the quadrilateral formed is a rhombus.
We know that,
A rhombus has all the sides equal and its diagonals bisect each other at $90^o$.
Hence, $\overline{AB}$ and $\overline{CD}$ are at right angles.
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