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

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


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  Metric relations for a tangent and a secant lines released from a point outside a circle  under the topic  Circles and their properties  of the section  Geometry  in this site.

Problem 1

The secant segment  PA  to a circle released from a point  P  outside                  
the circle has the measure of 9 units  (Figure 1).  Its external part  PB
has the measure of 4 units.

Find the measure of the tangent segment  PC  to the circle released from
the same point  P.

Solution

The tangent segment  PC  has the measure of  Geometric mean  of the
measures of the secant  PA  and its external part  PB  in accordance
with the lesson  Metric relations for a tangent and a secant lines
released from a point outside a circle
  under the topic  Circles and


      Figure 1.  To the Problem 1


their properties  of the section  Geometry  in this site:   |PC| = sqrt%28abs%28PA%29%2Aabs%28PB%29%29.

Hence,  |PC| = sqrt%289%2A4%29 = sqrt%2836%29 = 6.

Answer.  |PC| = 6.


Problem 2

The secant segment  PA  to a circle released from a point  P  outside                  
the circle has the measure of 16 units  (Figure 2).

Find the measure of the external part  PB  of the secant segment
if the tangent segment to the circle released from the same point  P 
has the measure of 12 units.

Solution

You can find the measure of the external part of the secant segment
PA  using metric relations for a tangent and a secant lines released
from a point outside a circle:


      Figure 2.  To the Problem 2

|PA|*|PB| = abs%28PC%29%5E2

(see the lesson  Metric relations for a tangent and a secant lines released from a point outside a circle  under the topic  Circles and their properties  of the section  Geometry  in this site).

It gives   |PB| = abs%28PC%29%5E2%2Fabs%28PA%29 = 12%5E2%2F16 = 144%2F16 = 9.

Answer.  |PB| = 9.


Problem 3

The secant segment  PA  to a circle released from a point  P  outside                  
the circle has the external part  PB  of 16 units long  (Figure 3).

Find the measure of the chord  AB  if the tangent segment  PC  to the
circle released from the same point  P  has the measure of 20 units.

Solution

First, you can find the measure of the secant segment  PA  using metric
relations for a tangent and a secant lines released from a point outside
a circle:


      Figure 3.  To the Problem 3

|PA|*|PB| = abs%28PC%29%5E2

(see the lesson  Metric relations for a tangent and a secant lines released from a point outside a circle  under the topic  Circles and their properties  of the section  Geometry  in this site).

It gives   |PA| = abs%28PC%29%5E2%2Fabs%28PB%29 = 20%5E2%2F16 = 400%2F16 = 25.

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

Answer.  |AB| = 9.


Problem 4

A secant  AP  and a tangent  CP  are released from the point  P             
outside the circle  (Figure 4).  The measure of the chord  AB
is  16  units;  the tangent  CP  has the measure of  15  units.

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

Solution

Let  x  be the measure of the segment  BP.
Then the measure of the secant  AP  is  x + 16.

Now apply the  Theorem  on metric relations for a tangent


        Figure 4.  To the Problem 4

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  under the topic  Circles and their properties  of the section  Geometry  in this site).

According to this  Theorem,  |AP|*|BP| = abs%28CP%29%5E2.
Substitute here  |AP| = x + 16  and  |BP| = x.  You get   (x + 16)*x = 15%5E2 = 225.
It gives the quadratic equation   x%5E2 + 16x - 225 = 0.
The roots of this equation are

x%5B1%5D=%28-16+%2B+sqrt%28%28-16%29%5E2+%2B+4%2A225%29%29%2F2=%28-16+%2B+sqrt%28256+%2B+900%29%29%2F2=%28-16+%2B+sqrt%281156%29%29%2F2=%28-16+%2B+34%29%2F2=9  and  x%5B2%5D=%28-16+-+sqrt%28%28-16%29%5E2+%2B+4%2A225%29%29%2F2=%28-16+-+sqrt%28256+%2B+900%29%29%2F2=%28-4+-+sqrt%281156%29%29%2F2=%28-16+-+34%29%2F2=-25.

Only the root  x%5B1%5D = 9  is the solution.  (The measure of the segment should be a positive real number).  Thus  |BP| = 9,  |AP| = 9 + 16 = 25.

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


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

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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