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

= 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

= 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

= 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