Lesson Solved problems on a tangent and a secant lines released from a point outside a circle

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<H2>Solved problems on a tangent and a secant lines released from a point outside a circle</H2>

In this lesson you will find some typical solved problems on a tangent and a secant lines released from a point outside a given circle.
The theoretical base for solving these problems is the lesson &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>&nbsp; under the topic &nbsp;Circles and their properties&nbsp; of the section &nbsp;Geometry&nbsp; in this site.

<H3>Problem 1</H3><TABLE> 
 <TR>
 <TD>The secant segment &nbsp;PA&nbsp; to a circle released from a point &nbsp;P&nbsp; outside&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

the circle has the measure of 9 units &nbsp;(Figure 1). &nbsp;Its external part &nbsp;PB

has the measure of 4 units.

Find the measure of the tangent segment &nbsp;PC&nbsp; to the circle released from

the same point &nbsp;P.

Solution

The tangent segment &nbsp;PC&nbsp; has the measure of &nbsp;Geometric mean&nbsp; of the

measures of the secant &nbsp;PA&nbsp; and its external part &nbsp;PB&nbsp; in accordance

with the lesson &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>&nbsp; under the topic &nbsp;Circles and

</TD>
 <TD>
{{{drawing( 320, 220, -5.5, 10.5, -5.5, 5.5, 
 circle(0.0, 0.0, 5, 5),

circle(0.0, 0.0, 0.08, 0.08),
            locate (-0.1, -0.1, O),

line( -3,   -4, 10,   2),
              locate (-3.3, -3.9, A),
              locate ( 5.1, -0.2, B),

locate ( 10.1,  2.5, P),

line( 1.60, 4.74, 10, 2), 
 locate ( 1.40, 5.5, C)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. &nbsp;To the Problem 1

</TD>
 </TR>
</TABLE>their properties&nbsp; of the section &nbsp;Geometry&nbsp; in this site: &nbsp;&nbsp;|PC| = {{{sqrt(abs(PA)*abs(PB))}}}.

Hence, &nbsp;|PC| = {{{sqrt(9*4)}}} = {{{sqrt(36)}}} = 6.

Answer. &nbsp;|PC| = 6.

<H3>Problem 2</H3><TABLE> 
 <TR>
 <TD>The secant segment &nbsp;PA&nbsp; to a circle released from a point &nbsp;P&nbsp; outside&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

the circle has the measure of 16 units &nbsp;(Figure 2).

Find the measure of the external part &nbsp;PB&nbsp; of the secant segment

if the tangent segment to the circle released from the same point &nbsp;P&nbsp;

has the measure of 12 units.

Solution

You can find the measure of the external part of the secant segment

PA&nbsp; using metric relations for a tangent and a secant lines released

from a point outside a circle:

</TD>
 <TD>
{{{drawing( 320, 220, -10.5, 5.5, -5.5, 5.5, 
 circle(0.0, 0.0, 5, 5),

circle(0.0, 0.0, 0.08, 0.08),
            locate (-0.1, -0.1, O),

line(  2,   -4.58, -10,   2),
              locate ( 1.9, -4.6, A),
              locate (-5.3, -0.7, B),

locate (-10.5,  2.5, P),

line( -1.60, 4.74, -10, 2), 
 locate (-1.40, 5.5, C)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 2. &nbsp;To the Problem 2

</TD>
 </TR>
</TABLE>|PA|*|PB| = {{{abs(PC)^2}}}

(see the lesson &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>&nbsp; under the topic &nbsp;Circles and their properties&nbsp; of the section &nbsp;Geometry&nbsp; in this site).

It gives &nbsp;&nbsp;|PB| = {{{abs(PC)^2/abs(PA)}}} = {{{12^2/16}}} = {{{144/16}}} = 9.

Answer. &nbsp;|PB| = 9.

<H3>Problem 3</H3><TABLE> 
 <TR>
 <TD>The secant segment &nbsp;PA&nbsp; to a circle released from a point &nbsp;P&nbsp; outside&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

the circle has the external part &nbsp;PB&nbsp; of 16 units long &nbsp;(Figure 3).

Find the measure of the chord &nbsp;AB&nbsp; if the tangent segment &nbsp;PC&nbsp; to the

circle released from the same point &nbsp;P&nbsp; has the measure of 20 units.

Solution

First, you can find the measure of the secant segment &nbsp;PA&nbsp; using metric

relations for a tangent and a secant lines released from a point outside

a circle:

</TD>
 <TD>
{{{drawing( 320, 220, -10.5, 5.5, -5.5, 5.5, 
 circle(0.0, 0.0, 5, 5),

circle(0.0, 0.0, 0.08, 0.08),
            locate (-0.1, -0.1, O),

line( -1,   -4.9, -10,   3),
              locate (-1.0, -4.9, A),
              locate (-5.2, -1.4, B),

locate (-10.5,  3.5, P),

line(-1.60,    4.74, -10,   3), 
              locate (-1.40,  5.5, C),

locate (-6.0, 4.7, 20),
 locate (-7.3, 1.5, 16),
 locate (-2.8, -2.5, x)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 3. &nbsp;To the Problem 3

</TD>
 </TR>
</TABLE>|PA|*|PB| = {{{abs(PC)^2}}}

It gives &nbsp;&nbsp;|PA| = {{{abs(PC)^2/abs(PB)}}} = {{{20^2/16}}} = {{{400/16}}} = 25.

Hence, &nbsp;&nbsp;|AB| = |PA| - |PB| = 25 - 16 = 9.

Answer. &nbsp;|AB| = 9.

<H3>Problem 4</H3><TABLE> 
 <TR>
 <TD>A secant &nbsp;AP&nbsp; and a tangent &nbsp;CP&nbsp; are released from the point &nbsp;P&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

outside the circle &nbsp;(Figure 4). &nbsp;The measure of the chord &nbsp;AB

is &nbsp;16&nbsp; units; &nbsp;the tangent &nbsp;CP&nbsp; has the measure of &nbsp;15&nbsp; units.

Find the measures of the secant &nbsp;AP&nbsp; and its external part &nbsp;BP.

Solution

Let &nbsp;x&nbsp; be the measure of the segment &nbsp;BP.

Then the measure of the secant &nbsp;AP&nbsp; is &nbsp;x + 16.

Now apply the &nbsp;Theorem&nbsp; on metric relations for a tangent

</TD>
 <TD>
{{{drawing( 320, 220, -5.5, 10.5, -5.5, 5.5, 
 circle(0.0, 0.0, 5, 5),

circle(0.0, 0.0, 0.08, 0.08),
            locate (-0.1, -0.1, O),

line( -3,   -4, 10,   2),
              locate (-3.3,  -3.9, A),
              locate ( 5.1, -0.2, B),

locate ( 10.1,  2.5, P),

line( 1.60,    4.74, 10,   2), 
              locate ( 1.40,  5.5, C),

locate ( 6.0, 4.1, 15),
 locate ( 0.2, -1.4, 16),
 locate ( 6.3, 1.3, x)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 4. &nbsp;To the Problem 4

</TD>
 </TR>
</TABLE>and a secant lines released from a point outside a circle &nbsp;(lesson &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>&nbsp; under the topic &nbsp;Circles and their properties&nbsp; of the section &nbspGeometry&nbsp; in this site).

According to this &nbsp;Theorem, &nbsp;|AP|*|BP| = {{{abs(CP)^2}}}.

Substitute here &nbsp;|AP| = x + 16 &nbsp;and&nbsp; |BP| = x. &nbsp;You get &nbsp;&nbsp;(x + 16)*x = {{{15^2}}} = 225.

It gives the quadratic equation &nbsp;&nbsp;{{{x^2}}} + {{{16x}}} - {{{225}}} = 0.

The roots of this equation are

{{{x[1]}}}={{{(-16 + sqrt((-16)^2 + 4*225))/2}}}={{{(-16 + sqrt(256 + 900))/2}}}={{{(-16 + sqrt(1156))/2}}}={{{(-16 + 34)/2}}}=9 &nbsp;and&nbsp; {{{x[2]}}}={{{(-16 - sqrt((-16)^2 + 4*225))/2}}}={{{(-16 - sqrt(256 + 900))/2}}}={{{(-4 - sqrt(1156))/2}}}={{{(-16 - 34)/2}}}=-25.

Only the root &nbsp;{{{x[1]}}} = {{{9}}}&nbsp; is the solution. &nbsp;(The measure of the segment should be a positive real number). &nbsp;Thus &nbsp;|BP| = 9, &nbsp;|AP| = 9 + 16 = 25.

Answer. &nbsp;The solution is &nbsp;|BP| = 9, &nbsp;|AP| = 25.

My other lessons on circles in this site, &nbsp;in the logical order, &nbsp;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/A-tangent-line-to-a-circle-is-perpendicular-to-the-radius-drawn-to-the-tangent-point.lesson>A tangent line to a circle is perpendicular to the radius drawn to the tangent point</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 topic &nbsp;Circles and their properties&nbsp; of the section &nbsp;Geometry, &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=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 current topic.

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>.