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Coursework Notes - Shape & Space





subtended angles



cyclic quadrilaterals



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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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.





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circles - tangents



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





the tangents subtend equal angles at the centre of the circle


angle POA = angle POB



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.




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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