<p>\(O\) denotes the centre. Distances from the centre to chords are measured perpendicularly.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 560px;"><colgroup><col style="width: 19.6429%;"><col style="width: 41.0714%;"><col style="width: 39.2857%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Property</strong></th><th style="text-align: center; --cell-justify: center"><strong>Statement and reason</strong></th><th style="text-align: center; --cell-justify: center"><strong>Diagram</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Perpendicular bisector of a chord</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="eyJheGlzIjpmYWxzZSwiZ3JpZCI6ZmFsc2UsIndpZHRoIjoxOTAsImhlaWdodCI6MTc1LCJlbGVtZW50cyI6W3siaWQiOiJ1MTYtdDMtY2cxLWMiLCJ0eXBlIjoiY2lyY2xlIiwiY29sb3IiOiIjMjU2M2ViIiwiY2VudGVyIjpbMCwwXSwicmFkaXVzIjoyfSx7ImlkIjoidTE2LXQzLWNnMS1hbSIsInR5cGUiOiJzZWdtZW50IiwicG9pbnQxIjpbLTEuNiwtMS4yXSwicG9pbnQyIjpbMCwtMS4yXSwibWlkZGxlQXJyb3ciOiJ0aWNrIiwibWlkZGxlTWFya2VyQ291bnQiOjF9LHsiaWQiOiJ1MTYtdDMtY2cxLW1iIiwidHlwZSI6InNlZ21lbnQiLCJwb2ludDEiOlswLC0xLjJdLCJwb2ludDIiOlsxLjYsLTEuMl0sIm1pZGRsZUFycm93IjoidGljayIsIm1pZGRsZU1hcmtlckNvdW50IjoxfSx7ImlkIjoidTE2LXQzLWNnMS1vbSIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwicG9pbnQxIjpbMCwwXSwicG9pbnQyIjpbMCwtMS4yXX0seyJpZCI6InUxNi10My1jZzEtcmEiLCJ0eXBlIjoiYW5nbGUiLCJjb2xvciI6IiNkYzI2MjYiLCJsYWJlbCI6IiIsInBvaW50MSI6WzEuNiwtMS4yXSwicG9pbnQyIjpbMCwtMS4yXSwicG9pbnQzIjpbMCwwXX0seyJ4IjowLCJ5IjowLCJpZCI6InUxNi10My1jZzEtcG8iLCJ0eXBlIjoicG9pbnQiLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJPIiwibGFiZWxPZmZzZXQiOls2LDhdfSx7IngiOi0xLjYsInkiOi0xLjIsImlkIjoidTE2LXQzLWNnMS1wYSIsInR5cGUiOiJwb2ludCIsImZpeGVkIjp0cnVlLCJsYWJlbCI6IkEiLCJsYWJlbE9mZnNldCI6Wy0xNCwtNF19LHsieCI6MS42LCJ5IjotMS4yLCJpZCI6InUxNi10My1jZzEtcGIiLCJ0eXBlIjoicG9pbnQiLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJCIiwibGFiZWxPZmZzZXQiOls4LC00XX0seyJ4IjowLCJ5IjotMS4yLCJpZCI6InUxNi10My1jZzEtcG0iLCJ0eXBlIjoicG9pbnQiLCJmaXhlZCI6dHJ1ZSwibGFiZWwiOiJNIiwibGFiZWxPZmZzZXQiOlstNCwtMTZdfV0sInNob3dYQXhpcyI6ZmFsc2UsInNob3dZQXhpcyI6ZmFsc2UsImJvdW5kaW5nQm94IjpbLTMsMi44LDMsLTNdLCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlfQ==" data-width="190" data-height="175" data-proportional style="width:190px;height:175px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:190px;--graph-aspect:190 / 175;"></div></td></tr><tr><td style="text-align: center; --cell-justify: center">Equal chords</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="190" data-height="175" data-proportional style="width:190px;height:175px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:190px;--graph-aspect:190 / 175;"></div></td></tr><tr><td style="text-align: center; --cell-justify: center">Tangent and radius</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="190" data-height="175" data-proportional style="width:190px;height:175px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:190px;--graph-aspect:190 / 175;"></div></td></tr><tr><td style="text-align: center; --cell-justify: center">Tangents from an external point</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><div class="graph-block" data-graph-config="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" data-width="210" data-height="155" data-proportional style="width:210px;height:155px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:210px;--graph-aspect:210 / 155;"></div></td></tr></tbody></table></div></div>

\(O\) denotes the centre. Distances from the centre to chords are measured perpendicularly.

The perpendicular from the centre bisects the chord, and the perpendicular bisector of a chord passes through the centre.
If \(OM\perp AB\), then \(AM=BM\).
(\(\perp\) bisector of chord)

Equal chords are equidistant from the centre.
If \(AB=CD\), then \(OM=ON\).
(equal chords, equidistant from centre)

A tangent is perpendicular to the radius drawn to the point of contact.
\(OA\perp PQ\)
(tangent \(\perp\) radius)

Tangents from an external point are equal in length, and the line to the centre bisects the angle between them.
\(PA=PB\)
(tangents from an external point)

Properties of Circles

Properties of Circles

Video Solution:

Solution:

Solution locked

Property	Statement and reason	Diagram
Perpendicular bisector of a chord
Equal chords
Tangent and radius
Tangents from an external point