Lesson Solved problems on secants that intersect outside a circle

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Solved problems on secants that intersect outside a circle

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 Metric relations for secants intersecting outside a circle under the topic Circles and their properties of the section Geometry in this site.

Problem 1

The secants AP and BP intersect at the point P outside the circle
(Figure 1). The measures of the three segments AP, CP and DP
are known; they are |AP| = 15, |CP| = 6 and |DP| = 5.

Find the measure of the secant BP.

Solution

Apply the Theorem on secants that intersect outside a circle
(lesson Metric relations for secants intersecting outside a circle
under the topic Circles and their properties of the section
Geometry in this site).

Figure 1. To the Problem 1

According to this Theorem, |AP|*|CP| = |BP|*|DP|.
Substitute the given data. You get 15*6 = |BP|*5.
It implies |BP| = 15%2A6%2F5

= 18.

Answer. |BP| = 18.

Problem 2

The secants AP and BP intersect at the point P outside the circle
(Figure 2). The measures of the three segments AC, CP and DP
are known; they are |AC| = 6, |CP| = 4 and |DP| = 5.

Find the measure of the chord BD.

Solution

First, let us find the measure of the full secant AP.
It is |AP| = |AC| + |CP| = 6 + 4 = 10.

Now apply the Theorem on secants that intersect outside a circle
(lesson Metric relations for secants intersecting outside a circle

Figure 2. To the Problem 2

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

According to this Theorem, |AP|*|CP| = |BP|*|DP|.
Substitute the known data. You get 10*4 = |BP|*5.
It implies |BP| = 10%2A4%2F5

= 8.

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

Answer. |BD| = 3.

Problem 3

The secants AP and BP intersect at the point P outside the circle
(Figure 3). The measure of the chord AC is 4 units; the chord BD
has the measure of 7 units and the segment DP has the measure of
5 units.

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

Solution

First, let us find the measure of the secant BP.
It is |BP| = |BD| + |DP| = 7 + 5 = 12.

Now, let x be the measure of the segment CP.
Then the measure of the secant AP is 4 + x.

Figure 3. To the Problem 3

Next, apply the Theorem on chords that intersect within a circle (lesson The parts of chords that intersect inside a circle under the topic Circles and their properties of the section Geometry in this site).

According to this Theorem, |AP|*|CP| = |BP|*|DP|.
Substitute here |AP| = 4 + x and |CP| = x. You get (x + 4)*x = 12*5.
It gives the quadratic equation x%5E2

= 0.
The roots of this equation are

x%5B1%5D

%28-4+%2B+sqrt%28%28-4%29%5E2+%2B+4%2A60%29%29%2F2

= 6 and

%28-4+-+sqrt%28%28-4%29%5E2+%2B+4%2A60%29%29%2F2

= -10.

Only the root x%5B1%5D

is the solution. (The measure of the segment should be a positive real number). Thus |CP| = 6, |AP| = 6 + 4 = 10.

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

Problem 4

If a secant line is released from a point outside a circle, then the product of the measures of the secant and its external part is equal to L%5E2+-+R%5E2

, where L is the distance from the given point outside the circle to the circle's center and R is the circle radius.
Or, which is the same, 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.

Solution

Let us consider a circle with the center at a point O (Figure 4a).
Let PA be a secant passing through the point P in the exterior of
the circle with A as the distant and C as the closest to P
intersection points at the circle. Let PC be the tangent line to the
circle passing through the same point P.

The Theorem states that

|PA|*|PC| = L%5E2+-+R%5E2

. (1)

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

Figure 4a. To the Problem 4

Figure 4b. To the solution of the Problem 4

and R is the circle radius.

For the proof, let us draw the secant PD passing through the point P and the center of the circle O (Figure 4b).
Let D be the distant and E be the closest to P intersection points of this secant with the circle.
Then we have |PA|*|PC| = |PD|*|PE| in accordance with the lesson Metric relations for secants intersecting outside a circle under the current topic.
You can rewrite the right side of the last equality in the form |PD|*|PE| = (|PO| + R)*(|PO| - R) = (L + R)*(L - R) = L%5E2

.

Thus the first part of the equality (1) is proved: |PA|*|PC| = L%5E2

.

Finally, the last expression L%5E2

is nothing else as the square of the measure of the tangent segment: L%5E2

(Figure 4b).
It follows the Pythagorean Theorem, since the radius drawn to the tangent point is perpendicular to the tangent line (see the lesson A tangent line to a circle is perpendicular to the radius drawn to the tangent point under the current topic).

This chain of arguments proves the equality (1).

Note
It was proved in the lesson Metric relations for secants intersecting outside a circle 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. 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: if the product of a secant and its external part has an invariant value, then what is this invariant value? What is its geometrical sense?

Now you know the answer: the product of the measures of a secant and its external part is equal to - , where L is the distance from the given point outside the circle to the center of the circle and R is the circle's radius. Or, it is the same as the square of the tangent segment to the circle released from the given point.

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 a tangent and a secant lines released from a point 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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