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shape & space : circles
 

parts

subtended angles

 

 


  • a drop-down menu of resources for 'circles'

    • The parts of a circle

    circle parts

    centre - the point within the circle where the distance to points on the circumference is the same.

    radius - the distance from the centre to any point on the circle. The diameter is twice the radius.

    circumference(perimeter) - the distance around a circle.

    chord is a straight line joining two points on the circumference.

    diameter - a chord(of max. length) passing through the centre

    sector - a region enclosed by two radii and an arc.

    segment - the region enclosed by a chord and an arc of the circle.

    tangent - a straight line making contact at one point on the circumference, such that the radius from the centre is at right angles to the line.

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    Subtended angles

    When a chord subtends an angle on the circumference of a circle, the angle subtended at the centre of the circle is twice the angle.

    circle - subtended angles #1

     

    A diameter subtends a right-angle at the circumference

    90 deg. subtended angle

    angle XPZ = 90 deg.

    Angles subtended by a chord onto the circumference of a circle are equal.

    subtended angles #2

    angle ADB = angle ACB

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    Chords

    chord #1

    The line joining the centre of a circle and the mid-point of a chord is perpendicular to the chord. The chord is bisected into two equal halves.

    XP = PY

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    Tangents

    circles - tangents

    The tangents to a circle from a point are equal in length.

    AP = BP

    also,

    the tangents subtend equal angles at the centre of the circle

    angle POA = angle POB

    and,

    the angles between the tangents and the line joining the centre of the circle and the point are equal.

    angle APO = angle BPO

    note : Triangle APO and triangle BPO are congruent.

     

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    The angle between a tangent and a chord

    The angle between a tangent and a chord is equal to the angle subtended by the chord in the opposite segment.

    circle - tangent #2

    angle ZPY = angle PXY

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    Cyclic quadrilaterals

    Opposite angles in a cyclic quadrilateral add up to 180 deg.

    circle - quads

    As with all quadrilaterals, the sum of the interior angles = 360 deg.

    Any exterior angle of a cyclic quadrilateral equals the interior opposite angle.

     

     

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