Lesson A tangent line to a circle is perpendicular to the radius drawn to the tangent point

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A tangent line to a circle is perpendicular to the radius drawn to the tangent point


Reminder
In the lesson  A circle,  its chords, tangent and secant lines - major definitions  a  tangent line to a circle  is defined as a straight line which has only one
common point with the circle.

In this lesson you will find the proofs to these statements:

    1)  a tangent line to a circle is perpendicular to the radius drawn to the tangent point,  and
    2)  a straight line perpendicular to a radius of a circle at its endpoint is a tangent line to the circle.

Theorem 1

A tangent line to a circle is perpendicular to the radius drawn to the tangent point

Proof

We are given a circle with the center  O  (Figure 1a)  and the tangent line  AB  to the circle.          
The tangent point is the point  A  of the circle.  The radius  OA  is drawn to the tangent point.

The Theorem states that the radius  OA  is perpendicular to the tangent line  AB,  or,
 in other words,  the angle  OAB  is the right angle.

For the proof,  let us assume that the angle  OAB  is not the right angle.
Then, considering two adjacent angles to the radius  OA,  we can choose the lesser of these
two.  So,  we can suppose that the angle  OAB  is an acute angle  (see the  Figure 2a).

Now,  let us draw the perpendicular  OC  from the point  O  to the straight line  AB
(it will be distinct from  OA,  due to the assumption).


Figure 1a.  To the Theorem 1      



Figure 1b.  To the proof
      of the  Theorem 1

Next, let us construct the straight segment  CD  in the straight line  AB  congruent to the straight line segment  AC,  and connect the points  O  and  D.
Then the triangles  DELTAOCA  and  DELTAOCD  are congruent as the right-angled triangles having congruent legs.  Hence,  the segments  OA  and  OD  are congruent.

This means that the point  D  belongs to the given circle,  because it lies at the same distance from the center  O  as the point  A.
Thus we constructed the point  D  in the straight line  AB  which belongs to the circle,  i.e.  is the second intersection point of the straight line  AB  and the circle.
It contradicts to the condition that the straight line  AB  is the tangent line to the circle.

The contradiction proves that the angle  OAB  is the right angle.

Theorem 2

A straight line perpendicular to a radius of a circle at its endpoint is a tangent line to the circle

Proof

We are given a circle with the center  O  (Figure 2a)  and a straight line  AB  which is perpen-         
dicular to the radius  OA  of the circle at its endpoint  A.

The Theorem states that the straight line  AB  is the tangent line to the circle.

For the proof,  let us assume that the straight line  AB  is not the tangent line to the circle.

The straight line  AB  just contains one common point with the given circle - it is the endpoint
A  of the radius  OA.  So,  our assumption means that the straight line  AB  has the second
common point with the circle.  Let us call this second common point as  D  (Figure 2b).


Figure 2a.  To the Theorem 2      



Figure 2b.  To the proof
      of the  Theorem 2

Consider the triangle  DELTAOAD.  It is the isosceles triangle,  because its sides  OA  and  OD  are of equal length as the radii of the circle.
Hence,  the angles  LOAD  and  LODA  are congruent as the angles at the base of the isosceles triangles.
Since the sum of the interior angles of the triangle is equal to  180°,  each of the angles  LOAD  and  LODA  is lesser than  90°.  This contradicts to the Theorem condition that the angle LOAD  is equal to  90°.

The contradiction proves that straight line  AB  is the tangent line to the circle.

Summary

A straight line is a tangent line to a circle if and only if it is perpendicular to the radius drawn to the tangent point.

The two definitions of the tangent line to a circle are equivalent:
    1)  a straight line is a tangent line to a circle if it has only one common point with the circle,  and
    2)  a straight line is a tangent line to a circle if it is perpendicular to the radius drawn to the tangent point.


My other lessons on circles in this site are
    - A circle, its chords, tangent and secant lines - the major definitions,
    - The longer is the chord the larger its central angle is,
    - The chords of a circle and the radii perpendicular to the chords,
    - An inscribed angle in a circle,
    - Two parallel secants to a circle cut off congruent arcs,
    - The angle between two chords intersecting inside a circle,
    - The angle between two secants intersecting outside a circle,
    - The angle between a chord and a tangent line to a circle,
    - Tangent segments to a circle from a point outside the circle,
    - The converse theorem on inscribed angles,
    - The parts of chords that intersect inside a circle,
    - Metric relations for secants intersecting outside a circle  and
    - Metric relations for a tangent and a secant lines released from a point outside a circle
under the current topic,  and
    - HOW TO bisect an arc in a circle using a compass and a ruler,
    - HOW TO find the center of a circle given by two chords,
    - Solved problems on a radius and a tangent line to a circle,
    - Solved problems on inscribed angles,
    - A property of the angles of a quadrilateral inscribed in a circle,
    - An isosceles trapezoid can be inscribed in a circle.
    - HOW TO construct a tangent line to a circle at a given point on the circle,
    - HOW TO construct a tangent line to a circle through a given point outside the circle,
    - HOW TO construct a common exterior tangent line to two circles,
    - HOW TO construct a common interior tangent line to two circles,
    - Solved problems on chords that intersect within a circle,
    - Solved problems on secants that intersect outside a circle,
    - Solved problems on a tangent and a secant lines released from a point outside a circle
    - The radius of a circle inscribed into a right angled triangle
    - Solved problems on tangent lines released from a point outside a circle
under the topic  Geometry  of the section  Word problems.

The overview of lessons on Properties of Circles is in this file  PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS.
You can use the overview file or the list of links above to navigate over these lessons.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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