View source for Tangents and Normals

[[File:CircleParts.png |300px|right]]
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===== Description=====

[https://en.wikipedia.org/wiki/Tangent '''Tangents'''] are lines ''just touching'' a given curve and its [https://en.wikipedia.org/wiki/Normal_%28geometry%29 '''Normal'''] is a line perpendicular to it at the point of contact ''(or point of '''Tangency''')''.


Taking the common case of a '''[[Circles|Circle]]''', the '''Normal''' to a '''Tangent''' from a point '''P''' on the circumference is a line joining the point to the circle centre - and the '''Tangent''' is at right angles to the '''Normal'''. This forms the basis for methods of constructing '''Tangents''' such that the '''Point of Tangency''' is given accurately.<div style="clear: both;"></div>
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===== To construct a Tangent to a Circle from a point outside its circumference: =====

[[File:TangentPointCircle.png |350px|right]]
* Join the point to the circle centre.
* Draw a semi-circle on this line as the diameter.
* The Point of Contact (''or Tangency)'' for the Tangent is where the semi-cicle intersects the circle and its Normal joins this point to the circle centre ''[[Circles|(angle in semi-circle)]]''.
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===== Construction of Tangents to an Ellipse =====

'''To construct a Tangent to an Ellipse from any point on the curve:'''
* Join the point to each of the two Foci as shown and bisect the angle found between the two lines.
* The bisector is the Normal and a Tangent may be drawn at right angles to it.
'''To construct a Tangent to an Ellipse from any point outside the curve:'''
* Strike an arc with the point as centre and radius equal to its distance from the Focus as shown.
* Strike a second arc with the second Focus as centre and radius set to equal the Major Axis.
* Join the points of intersection of the two arcs to the second Focus as shown.
* Where the two lines intersect the curve give the two Points of Tangency.
* Join the point to these Points of Tangency to produce the two possible Tangents and their Normals may be drawn at right angles to them.
{|
|-
| [[File:Tangent-Ellipse.png |350px]]
| [[File:Tangent-PointEllipse.png |350px]]
|}
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===== Construction of Tangents to a Parabola =====

'''Three methods to construct a Tangent to a point on the Parabola'''

'''Method 1.'''
* Join the point to the Focus and draw from it a perpendicular to the Directrix.
* The bisector of the two lines created is the Tangent and its Normal may be constructed at right angles to it.
'''Method 2.'''
* Draw a line from the point perpendicular to the Axis ''(i.e. an Ordinate)''
* With the Vertex as centre and radius the distance from the Vertex to where the Ordinate intersects the Axis, draw a semi-circle. 
* Draw the Tangent to the point from where the semi-circle cuts the Axis as shown and its Normal may be constructed at right angles to it.
'''Method 3.'''
* Join the point to the Focus construct a line at right angles to it from the Focus.
* Draw the Tangent to the point from the line's intersection the Directrix and its Normal may be constructed at right angles to it.
{|
|-
| [[File:Tangent-Parabola.png|250px]]
| [[File:Tangent-Parabola2.png|250px]]
| [[File:Tangent-Parabola3.png|250px]]
|}
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[[File:Tangent-PointParabola.png|200px|right]]

=====To construct a Tangent from a point outside the Parabola =====

* Join the point (P) to the Focus(F) and using this as diameter, decribe a circle.
* Draw Tangent at the Vertex (V) perpendicular to the Axis to intersect the circle circumference in X and Y.
* Join PX and PY and produce both to touch the Parabola generating the two Tangents as shown.
* To draw a Normal, produce a Tangent ''(e.g. YP)'' to meet the Directrix in D and join D to the Focus(F).
* A line drawn perpendicular to DF will intersect the Parabola at a Point of Tangency(T).
* Construct a perpendicular to the Tangent from its Point of Tangency(T) to create its Normal. 

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[[File:ConicsProperties.png |350px|right]]

===== Other terms common to all conics =====

'''Tangents''' and '''Normals''' are two terms used to describe properties of all conics. Other common properties are as follows:

* A '''Chord''' is a straight line joining two points on the curve.
* A '''Focal Chord''' is a Chord which passses through the '''Focus'''
* The mid-points of parallel Chords lie in a straight line called a '''Diameter'''
* A perpendicular from a point on the '''Axis''' is called an '''Ordinate''' and if it goes straight across to the other side, it is a '''Double Ordinate'''.
* The Double Ordinate through the Focus is the '''Latus Rectum''' ''(translates from Latin as ‘The Right Side’)''.
* A '''Normal''' to any conic is at right angles to a '''Tangent''' at the '''Point of Contact'''


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[[Category:Secondary]]
[[Category:Geometry]]