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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
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Figure 1. To the Problem 1
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their properties of the section Geometry in this site: |PC| = .
Hence, |PC| = = = 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:
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Figure 2. To the Problem 2
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|PA|*|PB| =
(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| = = = = 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:
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Figure 3. To the Problem 3
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|PA|*|PB| =
(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| = = = = 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
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Figure 4. To the Problem 4
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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| = .
Substitute here |AP| = x + 16 and |BP| = x. You get (x + 16)*x = = 225.
It gives the quadratic equation + - = 0.
The roots of this equation are
= = = = =9 and = = = = =-25.
Only the root = 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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