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

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

In this lesson you will find some solved problems on tangent lines released from a point outside a given circle.
The theoretical base for solving these problems is the lesson Tangent segments to a circle from a point outside the circle under the topic Circles and their properties of the section Geometry in this site.

Problem 1

The circle inscribed in a right-angled triangle with the legs

and

and the hypotenuse

has the radius %28a+%2B+b+-+c%29%2F2

. Prove.

Solution

Let DELTA

ABC be a right-angled triangle with the legs

and

and the hypotenuse

.
Consider the circle inscribed to the triangle DELTA

ABC (Figure 1), and let D, E and F
be the tangent points of the segments AC, BC and AB (the sides of the triangle) and
the circle. We have

= |CE| + |BE|,

= |CD| + |AD|,

= |BF| + |AF|.

Next, |BE| = |BF| and |AD| = |AF| in accordance with the lesson

Figure 1. To the Problem 1

Tangent segments to a circle from a point outside the circle under the topic Circles and their properties of the section Geometry in this site. Therefore,

a+%2B+b+-+c

= |CE| + |CD|. (1)

Now, consider the quadrilateral CDOE, where the point O is the center of the inscribed circle. This quadrilateral is a square, since OD and OE are perpendicular to the tangent segments CD and CE respectively (see the lesson A tangent line to a circle is perpendicular to the radius drawn to the tangent point in this site), and, hence, are parallel to the sides of the triangle AC and BC. Therefore, |CE| = |CD| =

, where

is the radius of the inscribed circle.

Then from (1) you get

. It is what has to be proved. The solution is completed.

Example 1

Find the radius of the inscribed circle into the right-angled triangle with the legs of 5 cm and 12 cm long.

Solution

First, let us calculate the hypotenuse of the right-angled triangle with the legs of a = 5 cm and b = 12 cm. It is

= 13 cm

in accordance with the Pythagorean Theorem.

Now, use the formula for the radius of the circle inscribed into the right-angled triangle. This formula was derived in the solution of the Problem 1 above.

= 2 cm.

Answer. The radius of the inscribed circle is 2 cm.

Example 2

Find the radius of the inscribed circle into the right-angled triangle with the leg of 8 cm and the hypotenuse of 17 cm long.

Solution

First, let us calculate the measure of the second leg the right-angled triangle which the leg of a = 8 cm and the hypotenuse of b = 17 cm. It is

= 15 cm

in accordance with the Pythagorean Theorem.

Now, use the formula for the radius of the circle inscribed into the right-angled triangle. This formula was derived in the solution of the Problem 1 above.

= 3 cm.

Answer. The radius of the inscribed circle is 3 cm.

Problem 2

If a quadrilateral is circumscribed about a circle, then the sums of its opposite sides are equal.

Solution

Let ABCD be a quadrilateral circumscribed about a circle (Figure 2),
and let E, F, G and H be the tangent points of the segments AB,
BC, CD and AD respectively (the sides of the quadrilateral)
and the circle. We have

|AB| = |AE| + |BE|,
|BC| = |BF| + |CF|,
|CD| = |CG| + |DG|,
|AD| = |AH| + |DH|.

From the other side,

Figure 2. To the Problem 2

|AE| = |AH|, |BE| = |BF|, |CF| = |CG| and |AH| = |DH|

in accordance with the lesson Tangent segments to a circle from a point outside the circle under the topic Circles and their properties of the section Geometry
in this site. Therefore,

|AB| + |CD| = |AE| + |BE| + |CG| + |DG| = |AH| + |BF| + |CF| + |AB| + |DH| = (|AH| + |DH|) + (|BF| + |CF|) = |AD| + |BC|.

Thus |AB| + |CD| = |AD| + |BC|. It is what has to be proved. The solution is completed.

Example 3

Find the measure of the fourth side of a quadrilateral circumscribed about a circle, if three other sides have the measures of 5 cm, 6 cm and 4 cm listed consecutively.

Solution

Let x be the measure of the fourth side of our quadrilateral. Since the quadrilateral is circumscribed about a circle, the sums of the measures of its opposite sides are equal in accordance with the Problem 2 above. Thus you can write the equation

5 + 4 = 6 + x.

From this equation, x = 5 + 4 - 6 = 3 cm.

Answer. The fourth side of the quadrilateral is of 3 cm long.

Example 4

A trapezoid is circumscribed about a circle. Find the measure of the mid-segment of a trapezoid, if its lateral sides are of 5 cm and 7 cm long.

Solution

Since the trapezoid is circumscribed about a circle, the sums of the measures of its opposite sides are equal in accordance with the Problem 2 above. Thus the sum
of the measures of its bases is equal to the sum of the measures of its lateral sides, i.e. 5 + 7 = 12 cm.
The mid-segment of a trapezoid has the measure half the sum of the measures of its bases (see the lesson Trapezoids and their mid-lines under the topic Polygons
of the section Geometry in this site. So, the mid-segment of our trapezoid has the measure of 12%2F2

= 6 cm.
Answer. The mid-segment of the trapezoid is of 6 cm long.

Example 5

The sides of a quadrilateral are of 5 cm, 6 cm, 7 sm and 8 cm long listed consecutively. Prove that this quadrilateral is not circumscribed about a circle.

Solution

If a quadrilateral is circumscribed about a circle, then the sums of its opposite sides are equal. In our case the sums of the opposite sides are of 5 + 7 = 12 cm
and 6 + 8 = 14 cm. Since the sums are not equal, the quadrilateral is not circumscribed about a circle.

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,
    - Solved problems on chords that intersect within a circle,
    - Solved problems on secants that intersect outside a circle and
    - 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
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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