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

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<H2>A tangent line to a circle is perpendicular to the radius drawn to the tangent point</H2>

<BLOCKQUOTE>Reminder

In the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Circles/A-circle-its-chords-tangent-and-secant-lines-the-major-definitions.lesson>A circle, &nbsp;its chords, tangent and secant lines - major definitions</A>&nbsp; a &nbsp;tangent line to a circle&nbsp; is defined as a straight line which has only one

common point with the circle.</BLOCKQUOTE>

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

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

<H3>Theorem 1</H3>A tangent line to a circle is perpendicular to the radius drawn to the tangent point

<TABLE> 
 <TR>
 <TD>
Proof

We are given a circle with the center &nbsp;O&nbsp; (Figure 1a)&nbsp; and the tangent line &nbsp;AB&nbsp; to the circle.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

The tangent point is the point &nbsp;A&nbsp; of the circle. &nbsp;The radius &nbsp;OA&nbsp; is drawn to the tangent point.

The Theorem states that the radius &nbsp;OA&nbsp; is perpendicular to the tangent line &nbsp;AB, &nbsp;or,

&nbsp;in other words,&nbsp; the angle &nbsp;OAB&nbsp; is the right angle.

For the proof, &nbsp;let us assume that the angle &nbsp;OAB&nbsp; is not the right angle.

Then, considering two adjacent angles to the radius &nbsp;OA, &nbsp;we can choose the lesser of these

two. &nbsp;So, &nbsp;we can suppose that the angle &nbsp;OAB&nbsp; is an acute angle &nbsp;(see the &nbsp;Figure 2a).

Now, &nbsp;let us draw the perpendicular &nbsp;OC&nbsp; from the point &nbsp;O&nbsp; to the straight line &nbsp;AB

(it will be distinct from &nbsp;OA, &nbsp;due to the assumption).

</TD>
 <TD>
{{{drawing( 180, 180, -6.0, 6.0, -6.0, 6.0, 
 circle( 0.0, 0.0, 5.0), 
 line( 0.0, 0.0, 0.0, 5.0),
 line( -5.0, 5.0, 5.0, 5.0),

locate(-0.25, 0.0, O),
            locate(-0.1,  5.99, A),
            locate( 4.5,  5.99, B),

arc(0, 5, 1.9, 1.9, 0, 90),

arc(0, 5, 1.9, 1.9, 90, 180)
)}}}

Figure 1a. &nbsp;To the Theorem 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</TD>
 <TD>
{{{drawing( 180, 180, -6.0, 6.0, -6.0, 6.0, 
 circle( 0.0, 0.0, 5.0), 
 line( 0.0, 0.0, 0.0, 5.0),
 line( -5.0, 5.8, 5.0, 3.8),

locate(-0.25, 0.0, O),
            locate(-0.1,  5.99, A),
            locate( 5.0,  4.99, B),

arc(0, 5, 1.5, 1.5, 20, 90),

green(line(0, 0, 1.1, 4.7)),
            locate( 0.9, 5.7, C),

green(line(0, 0, 2.0, 4.4)),
            locate( 1.8, 5.4, D)
)}}}

Figure 1b. &nbsp;To the proof 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the &nbsp;Theorem 1

</TD>
 </TR>
</TABLE>

Next, let us construct the straight segment &nbsp;CD&nbsp; in the straight line &nbsp;AB&nbsp; congruent to the straight line segment &nbsp;AC, &nbsp;and connect the points &nbsp;O&nbsp; and &nbsp;D.

Then the triangles &nbsp;{{{DELTA}}}OCA&nbsp; and &nbsp;{{{DELTA}}}OCD&nbsp; are congruent as the right-angled triangles having congruent legs. &nbsp;Hence, &nbsp;the segments &nbsp;OA&nbsp; and &nbsp;OD&nbsp; are congruent.

This means that the point &nbsp;D&nbsp; belongs to the given circle, &nbsp;because it lies at the same distance from the center &nbsp;O&nbsp; as the point &nbsp;A.

Thus we constructed the point &nbsp;D&nbsp; in the straight line &nbsp;AB&nbsp; which belongs to the circle, &nbsp;i.e.&nbsp; is the second intersection point of the straight line &nbsp;AB&nbsp; and the circle.

It contradicts to the condition that the straight line &nbsp;AB&nbsp; is the tangent line to the circle.

The contradiction proves that the angle &nbsp;OAB&nbsp; is the right angle.

<H3>Theorem 2</H3>A straight line perpendicular to a radius of a circle at its endpoint is a tangent line to the circle

<TABLE> 
 <TR>
 <TD>
Proof

We are given a circle with the center &nbsp;O&nbsp; (Figure 2a)&nbsp; and a straight line &nbsp;AB&nbsp; which is perpen-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

dicular to the radius &nbsp;OA&nbsp; of the circle at its endpoint &nbsp;A.

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

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

The straight line &nbsp;AB&nbsp; just contains one common point with the given circle - it is the endpoint

A&nbsp; of the radius &nbsp;OA. &nbsp;So, &nbsp;our assumption means that the straight line &nbsp;AB&nbsp; has the second

common point with the circle. &nbsp;Let us call this second common point as &nbsp;D&nbsp; (Figure 2b).

</TD>
 <TD>
{{{drawing( 180, 180, -6.0, 6.0, -6.0, 6.0, 
 circle( 0.0, 0.0, 5.0), 
 line( 0.0, 0.0, 0.0, 5.0),
 line( -5.0, 5.0, 5.0, 5.0),

locate(-0.25, 0.0, O),
            locate(-0.1,  5.99, A),
            locate( 4.5,  5.99, B),

arc(0, 5, 1.9, 1.9, 0, 90),

arc(0, 5, 1.9, 1.9, 90, 180)
)}}}

Figure 2a. &nbsp;To the Theorem 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</TD>
 <TD>
{{{drawing( 180, 180, -6.0, 6.0, -6.0, 6.0, 
 circle( 0.0, 0.0, 5.0), 
 line( 0.0, 0.0, 0.0, 5.0),
 line( -5.0, 5.8, 5.0, 3.8),

locate(-0.25, 0.0, O),
            locate(-0.1,  5.99, A),
            locate( 5.0,  4.99, B),

arc(0, 5, 1.6, 1.6, 20, 90),

green(line(0, 0, 2.0, 4.4)),
            locate( 1.8, 5.4, D),

arc(2, 4.4, 1.5, 1.5, 115, 198)
)}}}

Figure 2b. &nbsp;To the proof 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the &nbsp;Theorem 2

</TD>
 </TR>
</TABLE>

Consider the triangle &nbsp;{{{DELTA}}}OAD. &nbsp;It is the isosceles triangle, &nbsp;because its sides &nbsp;OA&nbsp; and &nbsp;OD&nbsp; are of equal length as the radii of the circle.

Hence, &nbsp;the angles &nbsp;LOAD&nbsp; and &nbsp;LODA&nbsp; 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 &nbsp;180�, &nbsp;each of the angles &nbsp;LOAD&nbsp; and &nbsp;LODA&nbsp; is lesser than &nbsp;90�. &nbsp;This contradicts to the Theorem condition that the angle LOAD&nbsp; is equal to &nbsp;90�.

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

<H3>Summary</H3>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:

&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;a straight line is a tangent line to a circle if it has only one common point with the circle, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;2) &nbsp;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

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/A-circle-its-chords-tangent-and-secant-lines-the-major-definitions.lesson>A circle, its chords, tangent and secant lines - the major definitions</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-longer-is-the-chord-the-larger-its-central-angle-is.lesson>The longer is the chord the larger its central angle is</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-chords-in-a-circle-and-the-radii-perpendicular-to-the-chords.lesson>The chords of a circle and the radii perpendicular to the chords</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/An-inscribed-angle.lesson>An inscribed angle in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Two-parallel-secants-to-a-circle-cut-off-congruent-arcs.lesson>Two parallel secants to a circle cut off congruent arcs</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-two-chords-intersecting-inside-a-circle.lesson>The angle between two chords intersecting inside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-two-secants-intersecting-outside-a-circle.lesson>The angle between two secants intersecting outside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-a-chord-and-a-tangent-line-to-a-circle.lesson>The angle between a chord and a tangent line to a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Tangent-segments-to-a-circle-from-a-point-outside-the-circle.lesson>Tangent segments to a circle from a point outside the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-converse-theorem-on-inscribed-angles.lesson>The converse theorem on inscribed angles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-parts-of-chords-intersecting-inside-a-circle.lesson>The parts of chords that intersect inside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-metric-relations-for-secants-intersecting-outside-a-circle.lesson>Metric relations for secants intersecting outside a circle</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Metric-relations-for-a-tangent-and-a-secant-lines-released-from-a-point-outside-a-circle.lesson>Metric relations for a tangent and a secant lines released from a point outside a circle</A>

under the current topic, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-bisect-an-arc-in-a-circle-using-a-compass-and-a-ruler.lesson>HOW TO bisect an arc in a circle using a compass and a ruler</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-find-the-center-of-a-circle-given-by-two-chords.lesson>HOW TO find the center of a circle given by two chords</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-a-radius-and-a-tangent-line-to-a-circle.lesson>Solved problems on a radius and a tangent line to a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-inscribed-angles.lesson>Solved problems on inscribed angles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-property-of-the-angles-of-a-quadrilateral-inscribed-in-a-circle.lesson>A property of the angles of a quadrilateral inscribed in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Isosceles-trapezoid-can-be-inscribed-in-a-circle.lesson>An isosceles trapezoid can be inscribed in a circle</A>.

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-tangent-line-to-a-circle.lesson>HOW TO construct a tangent line to a circle at a given point on the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-tangent-line-to-a-circle-through-a-given-point-outside-the-circle.lesson>HOW TO construct a tangent line to a circle through a given point outside the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-common-exterior-tangent-line-to-two-circles.lesson>HOW TO construct a common exterior tangent line to two circles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-common-interior-tangent-line-to-two-circles.lesson>HOW TO construct a common interior tangent line to two circles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-chords-that-intersect-within-a-circle.lesson>Solved problems on chords that intersect within a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-secants-that-intersect-outside-a-circle.lesson>Solved problems on secants that intersect outside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-a-tangent-and-a-secant-lines-released-from-a-point-outside-a-circle.lesson>Solved problems on a tangent and a secant lines released from a point outside a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-radius-of-a-circle-inscribed-into-a-right-angled-triangle.lesson>The radius of a circle inscribed into a right angled triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-tangent-lines-released-from-a-point-outside-a-circle.lesson>Solved problems on tangent lines released from a point outside a circle</A>

under the topic &nbsp;Geometry&nbsp; of the section &nbsp;Word problems.

The overview of lessons on Properties of Circles is in this file &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Circles/PROPERTIES-OF-CIRCLES-THEIR-CHORDS-SECANTS-AND-TANGENTS.lesson>PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS</A>.
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 &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.