Lesson Solved problems on secants that intersect outside a circle

Source code of 'Solved problems on secants that intersect outside a circle'

This Lesson (Solved problems on secants that intersect outside a circle) was created by by ikleyn(52781)

: View Source, Show
About ikleyn:

<H2>Solved problems on secants that intersect outside a circle</H2>

In this lesson you will find some typical solved problems on secants intersecting outside a given circle.
The theoretical base for solving these problems is the lesson &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; 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 secants &nbsp;AP&nbsp; and &nbsp;BP&nbsp; intersect at the point &nbsp;P&nbsp; outside the circle&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

(Figure 1). &nbsp;The measures of the three segments &nbsp;AP, &nbsp;CP&nbsp; and &nbsp;DP

are known; &nbsp;they are &nbsp;|AP| = 15, &nbsp;|CP| = 6 &nbsp;and &nbsp;|DP| = 5.

Find the measure of the secant &nbsp;BP.

Solution

Apply the &nbsp;Theorem&nbsp; on secants that intersect outside a circle

(lesson <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>

under the topic &nbsp;Circles and their properties&nbsp; of the section

Geometry &nbsp;in this site).

</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 (-3.5, -4.0, A),
              locate ( 5.1, -0.2, C),

locate ( 10.1,  2.5, P),

line( -4, 3, 10, 2), 
 locate (-4.5, 3.5, B),
 locate ( 4.4, 3.3, D)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. &nbsp;To the Problem 1

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

According to this &nbsp;Theorem, &nbsp;|AP|*|CP| = |BP|*|DP|.

Substitute the given data. &nbsp;You get &nbsp;&nbsp;15*6 = |BP|*5. 
It implies &nbsp;|BP| = {{{15*6/5}}} = {{{90/5}}} = 18.

Answer. &nbsp;|BP| = 18.

<H3>Problem 2</H3><TABLE> 
 <TR>
 <TD>The secants &nbsp;AP&nbsp; and &nbsp;BP&nbsp; intersect at the point &nbsp;P&nbsp; outside the circle&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

(Figure 2). &nbsp;The measures of the three segments &nbsp;AC, &nbsp;CP&nbsp; and &nbsp;DP

are known; &nbsp;they are &nbsp;|AC| = 6, &nbsp;|CP| = 4 &nbsp;and &nbsp;|DP| = 5.

Find the measure of the chord &nbsp;BD.

Solution

First, &nbsp;let us find the measure of the full secant &nbsp;AP.

It is &nbsp;|AP| = |AC| + |CP| = 6 + 4 = 10.

Now apply the &nbsp;Theorem&nbsp; on secants that intersect outside a circle

(lesson &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>

</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 (-3.5, -4.0, A),
              locate ( 5.1, -0.2, C),

locate ( 10.1,  2.5, P),

line( -2, 4.58, 10, 2), 
 locate (-2.2, 5.4, B),
 locate ( 3.6, 4.2, D)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;Figure 2. &nbsp;To the Problem 2

</TD>
 </TR>
</TABLE>under the topic &nbsp;Circles and their properties&nbsp; of the section &nbsp;Geometry&nbsp; in this site).

According to this &nbsp;Theorem, &nbsp;|AP|*|CP| = |BP|*|DP|.

Substitute the known data. &nbsp;You get &nbsp;&nbsp;10*4 = |BP|*5. 
It implies &nbsp;|BP| = {{{10*4/5}}} = {{{40/5}}} = 8.

Hence, &nbsp;|BD| = |BP| - |DP| = 8 - 5 = 3.

Answer. &nbsp;|BD| = 3.

<H3>Problem 3</H3><TABLE> 
 <TR>
 <TD>The secants &nbsp;AP&nbsp; and &nbsp;BP&nbsp; intersect at the point &nbsp;P&nbsp; outside the circle&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

(Figure 3). &nbsp;The measure of the chord &nbsp;AC&nbsp; is &nbsp;4&nbsp; units; the chord &nbsp;BD&nbsp;

has the measure of &nbsp;7&nbsp; units and the segment &nbsp;DP&nbsp; has the measure of

5&nbsp; units.

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

Solution

First, &nbsp;let us find the measure of the secant &nbsp;BP.

It is &nbsp;|BP| = |BD| + |DP| = 7 + 5 = 12.

Now, &nbsp;let &nbsp;x&nbsp; be the measure of the segment &nbsp;CP.

Then the measure of the secant &nbsp;AP&nbsp; is &nbsp;4 + x.

</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(  0,   -5, 10,   2),
              locate (-0.2, -5.0, A),
              locate ( 4.7, -1.6, C),

locate ( 10.1,  2.5, P),

line( -4,    3, 10,   2), 
              locate (-4.5,  3.5, B),
              locate ( 4.4,  3.3, D),

locate (  0.0,  3.5, 7),
              locate (  7.0,  3.0, 5),
              locate (  2.0, -2.4, 4),

locate (  6.8,  0.8, x)

)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 3. &nbsp;To the Problem 3
 </TD>
 </TR>
</TABLE>Next, &nbsp;apply the &nbsp;Theorem&nbsp; on chords that intersect within a circle &nbsp;(lesson &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; under the topic &nbsp;Circles and their properties&nbsp; of the section &nbspGeometry&nbsp; in this site).

According to this &nbsp;Theorem, &nbsp;|AP|*|CP| = |BP|*|DP|.

Substitute here &nbsp;|AP| = 4 + x &nbsp;and&nbsp; |CP| = x. &nbsp;You get &nbsp;&nbsp;(x + 4)*x = 12*5.

It gives the quadratic equation &nbsp;&nbsp;{{{x^2}}} + {{{4x}}} - {{{60}}} = 0.

The roots of this equation are

{{{x[1]}}} = {{{(-4 + sqrt((-4)^2 + 4*60))/2}}} = {{{(-4 + sqrt(16 + 240))/2}}} = {{{(-4 + sqrt(256))/2}}} = {{{(-4 + 16)/2}}} = 6 &nbsp;&nbsp;and&nbsp;&nbsp; {{{x[2]}}} = {{{(-4 - sqrt((-4)^2 + 4*60))/2}}} = {{{(-4 - sqrt(16 + 240))/2}}} = {{{(-4 - sqrt(256))/2}}} = {{{(-4 - 16)/2}}} = -10.

Only the root &nbsp;{{{x[1]}}} = {{{6}}}&nbsp; is the solution. &nbsp;(The measure of the segment should be a positive real number). &nbsp;Thus &nbsp;|CP| = 6, &nbsp;|AP| = 6 + 4 = 10.

Answer. &nbsp;The solution is &nbsp;|CP| = 6, &nbsp;|AP| = 10.

<H3>Problem 4</H3>If a secant line is released from a point outside a circle, &nbsp;then the product of the measures of the secant and its external part is equal to &nbsp;{{{L^2 - R^2}}}, &nbsp;where &nbsp;L&nbsp; is the distance from the given point outside the circle to the circle's center and &nbsp;R&nbsp; is the circle radius.

Or, &nbsp;which is the same, &nbsp;the product of the measures of the secant and its external part is equal to the square of the tangent segment released from the given point.

<TABLE> 
 <TR>
 <TD>
Solution

Let us consider a circle with the center at a point &nbsp;O&nbsp; (Figure 4a).&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

Let &nbsp;PA&nbsp; be a secant passing through the point &nbsp;P&nbsp; in the exterior of

the circle with &nbsp;A&nbsp; as the distant and &nbsp;C&nbsp; as the closest to &nbsp;P

intersection points at the circle. &nbsp;Let &nbsp;PC&nbsp; be the tangent line to the

circle passing through the same point &nbsp;P.

The &nbsp;Theorem&nbsp; states that

|PA|*|PC| = {{{L^2 - R^2}}} = {{{abs(PB)^2}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

where &nbsp;L&nbsp; is the distance from the point &nbsp;P&nbsp; to the circle's center

</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 (-3.5, -4.0, A),
              locate ( 5.1, -0.2, C),

locate ( 10.1,  2.5, P),

line( 1.60, 4.74, 10, 2), 
 locate ( 1.40, 5.5, B)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;Figure 4a. &nbsp;To the Problem 4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</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 (-3.5, -4.0, A),
              locate ( 5.1, -0.2, C),

locate ( 10.1,  2.5, P),

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

green(line( -5,  -1,  10,  2)),

locate ( -5.4,  -0.7, D),
              locate (  4.9,   1.8, E),

blue(line( 0, 0, 1.60, 4.74))
)}}}
Figure 4b. &nbsp;To the solution of the Problem 4

</TD>
 </TR>
</TABLE>and &nbsp;R&nbsp; is the circle radius.

For the proof, &nbsp;let us draw the secant &nbsp;PD&nbsp; passing through the point &nbsp;P&nbsp; and the center of the circle &nbsp;O&nbsp; (Figure 4b).

Let &nbsp;D&nbsp; be the distant and &nbsp;E&nbsp; be the closest to &nbsp;P intersection points of this secant with the circle.

Then we have &nbsp;&nbsp;|PA|*|PC| = |PD|*|PE| &nbsp;&nbsp;in accordance with the lesson &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; under the current topic.

You can rewrite the right side of the last equality in the form &nbsp;&nbsp;|PD|*|PE| = (|PO| + R)*(|PO| - R) = (L + R)*(L - R) = {{{L^2}}} - {{{R^2}}}.

Thus the first part of the equality &nbsp;(1)&nbsp; is proved: &nbsp;&nbsp;|PA|*|PC| = {{{L^2}}} - {{{R^2}}}.

Finally, &nbsp;the last expression &nbsp;{{{L^2}}} - {{{R^2}}}&nbsp; is nothing else as the square of the measure of the tangent segment: &nbsp;&nbsp;{{{L^2}}} - {{{R^2}}} = {{{abs(PO)^2}}} - {{{abs(OB)^2}}} = {{{abs(PB)^2}}} &nbsp;(Figure 4b).

It follows the &nbsp;Pythagorean Theorem, &nbsp;since the radius drawn to the tangent point is perpendicular to the tangent line &nbsp;(see the lesson &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; under the current topic).

This chain of arguments proves the equality &nbsp;(1).

<BLOCKQUOTE><H3>Note</H3>It was proved in the lesson &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; that the product of the measures of a secant and its external part is the same for all secants passing through the given point outside the circle. &nbsp;It means that the product of a secant and its external part has the invariant value for all secants passing through the given point.

You may ask: &nbsp;if the product of a secant and its external part has an invariant value, &nbsp;then what is this invariant value? &nbsp;What is its geometrical sense?

Now you know the answer: &nbsp;the product of the measures of a secant and its external part is equal to &nbsp;{{{L^2}}} - {{{R^2}}}, &nbsp;where &nbsp;L&nbsp; is the distance from the given point outside the circle to the center of the circle and &nbsp;R&nbsp; is the circle's radius. &nbsp;Or, &nbsp;it is the same as the &nbsp;square of the tangent segment to the circle released from the given point.</BLOCKQUOTE>

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-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 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>.