We start by drawing the figure above.
The distance to the horizon is given by the line segment .
Let us convert the height of the mountain from feet into miles.
Since is tangent to the Earth, is a right triangle and we can use the Pythagorean Theorem.
Converse of a Tangent to a Circle
Converse of a Tangent to a Circle Theorem
If a line is perpendicular to the radius of a circle at its outer endpoint, then the line is tangent to the circle.
Proof.
We will prove this theorem by contradiction. Since the line is perpendicular to the radius at its outer endpoint it must touch the circle at point . For this line to be tangent to the circle, it must only touch the circle at this point and no other.
Assume that the line also intersects the circle at point .
Since both and are radii of the circle, , and is isosceles.
This means that .
It is impossible to have two right angles in the same triangle.
We arrived at a contradiction so our assumption must be incorrect. We conclude that line is tangent to the circle at point .
Example 3
Determine whether is tangent to the circle.
is tangent to the circle if .
To show that is a right triangle we use the Converse of the Pythagorean Theorem:
The lengths of the sides of the triangle satisfy the Pythagorean Theorem, so is perpendicular to and is therefore tangent to the circle.
Tangent Segments from a Common External Point
Tangent Segments from a Common External Point Theorem
If two segments from the same exterior point are tangent to the circle, then they are congruent.
Proof.
The figure above shows a diagram of the situation.
Given: is a tangent to the circle and is a tangent to the circle
Prove:
Statement Reason
1. is tangent to the circle 1. Given
2. 2. Tangent to a Circle Theorem
3. is a tangent to the circle 3. Given
4. 4. Tangent to a Circle Theorem
5. 5. Radii of same the circle
6. 6. Same line
7. 7. Hypotenuse-Leg congruence
8. 8. Congruent Parts of Congruent Triangles are Congruent
Example 4
Find the perimeter of the triangle.
All sides of the triangle are tangent to the circle.
The Tangent Segments from a Common External Point Theorem tells us that:
Example 5
An isosceles right triangle is circumscribed about a circle with diameter of . Find the hypotenuse of the triangle.
Let’s start by making a sketch.
Since and are radii of the circle and and are tangents to the circle,
and .
Therefore, quadrilateral is a square.
Therefore,
We can find the length of side by using the Pythagorean Theorem.
and are both isosceles right triangles, therefore and all the corresponding sides are proportional:
We can find the length of by using one of the ratios above:
Cross-multiply to obtain:
Rationalize the denominators:
The length of the hypotenuse is
Corollary to Tangent Segments Theorem
A line segment from an external point to the center of a circle bisects the angle formed by the tangent segments starting at that same external point.
Proof.
Given: is a tangent to the circle
is a tangent to the circle
is the center of the circle
Prove
Proof.
We will use a similar figure to the one we used we used to prove the tangent segments theorem (pictured above).
by congruence.
Therefore, .
Example 6
Show that the line is tangent to the circle . Find an equation for the line perpendicular to the tangent line at the point of tangency. Show that this line goes through the center of the circle.
To check that the line is tangent to the circle, substitute the equation of the line into the equation for the circle.
This has a double root at . This means that the line intersects the circle at only one point .
A perpendicular line to the tangent line would have a slope that is the negative reciprocal of the tangent line or .
The equation of the line can be written: .
We find the value of by plugging in the tangency point:
The equation is and we know that it passes through the origin since the intercept is zero.
This means that the radius of the circle is perpendicular to the tangent to the circle.
Lesson Summary
In this section we learned about tangents and their relationship to the circle. We found that a tangent line touches the circle at one point, which is the endpoint of a radius of the circle. The radius and tangent line are perpendicular to each other. We found out that if two segments are tangent to a circle, and share a common endpoint outside the circle; the segments are congruent.
Review Questions
Determine whether each segment is tangent to the given circle:
Find the measure of angle .
Find the missing length:
Find the values of the missing variables
Find the perimeter of the pentagon:
Find the perimeter of the parallelogram:
Find the perimeter of the right triangle:
Find the perimeter of the polygon:
Draw the line and the circle . Show that these graphs touch at only one point.
Find the slope of the segment that joins this point to the center of the circle, and compare your answer with the slope of the line .
Review Answers
Yes
Yes
No
solve for to obtain double root .
The slope of the line from to , which is the negative reciprocal of the slope of the line.
This means that the tangent line and radius are perpendicular.
Common Tangents and Tangent Circles
Learning Objectives
Solve problems involving common internal tangents of circles.
Solve problems involving common external tangents of circles.
Solve problems involving externally tangent circles.
Solve problems involving internally tangent circles.
Common tangents to two circles may be internal or external. A common internal tangent intersects the line segment connecting the centers of the two circles whereas a common external tangent does not.
Common External Tangents
Here is an example in which you might encounter the use of common external tangents.
Example 1
Find the distance between the centers of the circles in the figure.
Let’s label the diagram and draw a line segment that joins the centers of the two circles. Also draw the segment perpendicular the radius
Since is tangent to both circles, is perpendicular to both radii: and .
We can see that is a rectangle, therefore .
This means that .
is a right triangle with and . We can apply the Pythagorean Theorem to find the missing side, .
The distance between the centers is approximately
Common Internal Tangents
Here is an example in which you might encounter the use of common internal tangents.
Example 2
is tangent to both circles. Find the value of and the distance between the centers of the circles.
Therefore,
Using the Pythagorean Theorem on
Using the Pythagorean Theorem on
The distance between the centers of the circles is
Two circles are tangent to each other if they have only one common point. Two circles that have two common points are said to intersect each other.
Two circles can
be externally tangent if the circles are situated outside one another and internally tangent if one of them is situated inside the other.
Externally Tangent Circles
Here are some examples involving externally tangent circles.
Example 3
Circles tangent at are centered at and . Line is tangent to both circles at . Find the radius of the smaller circle if
tangent is perpendicular to the radius.
tangent is perpendicular to the radius.
In the right triangle .
We are also given that .
Therefore,
Also,
Therefore, by the similarity postulate.
The radius of the smaller circle is approximately .
Example 4
Two circles that are externally tangent have radii of and respectively. Find the length of tangent .
Label the figure as shown.
In and .
Therefore,
tangent is perpendicular to the radius.
tangent is perpendicular to the radius.
Therefore,
by the similarity postulate.
Therefore,
Internally Tangent Circles
Here is an example involving internally tangent circles.
Example 5
Two diameters of a circle of radius are drawn to make a central angle of . A smaller circle is placed inside the bigger circle so that it is tangent to the bigger circle and to both diameters. What is the radius of the smaller circle?
and are two tangents to the smaller circle from a common point so by Theorem 9-3, bisects
In we use
Draw from the points of tangency between the circles perpendicular to .
In we use
We also have because a tangent is perpendicular to the radius.
Therefore,
both equal
same angle.
Therefore, by the similarity postulate.
This gives us the ratio
( since they are both radii of the small circle).
Lesson Summary
In this section we learned about externally and internally tangent circles. We looked at the different cases when two circles are both tangent to the same line, and/or tangent to each other.
Review Questions
is tangent to both circles.
. Find .
and . Find .
and . Find .
and Find .
is tangent to both circles. Find the measure of angle .
and
and
For 9 and 10, find .
;
;
Circles tangent at are centered at and . is tangent to both circles at . Find the radius of the smaller circle if .
Four identical coins are lined up in a row as shown. The distance between the centers of the first and the fourth coin is . What is the radius of one of the coins?
Four circles are arranged inside an equilateral triangle as shown. If the triangle has sides equal to , what is the radius of the bigger circle? What are the radii of the smaller circles?
In the following drawing, each segment is tangent to each circle. The largest circle has a radius of The medium circle has a radius of What is the radius of the smallest circle tangent to the medium circle?
Circles centered at and are tangent at . Show that , and are collinear.
is a common external tangent to the two circles. is tangent to both circles. Prove that
A circle with a radius is centered at , and a circle with a radius is centered at , where and are apart. The common external tangent touches the small circle at and the large circle at . What kind of quadrilateral is What are the lengths of its sides?
Review Answers
Proof
Proof
Right trapezoid;
Arc Measures
Learning Objectives
Measure central angles and arcs of circles.
Find relationships between adjacent arcs.
Find relationships between arcs and chords.
Arc, Central Angle
In a circle, the central angle is formed by two radii of the circle with its vertex at the center of the circle. An arc is a section of the circle.
Minor and Major Arcs, Semicircle
A semicircle is half a circle. A major arc is longer than a semicircle and a minor arc is shorter than a semicircle.
An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this lesson we will concentrate on degree measure. The measure of the minor arc is the same as the measure of the central angle that corresponds to it. The measure of the major arc equals to minus the measure of the minor arc.
Minor arcs are named with two letters—the letters that denote the endpoints of the arc. In the figure above, the minor arc corresponding to the central angle is called . In order to prevent confusion, major arcs are named with three letters—the letters that denote the endpoints of the arc and any other point on the major arc. In the figure, the major arc corresponding to the central angle is called .
Congruent Arcs
Two arcs that correspond to congruent central angles will also be congruent. In the figure below, because they are vertical angles. This also means that .
Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.
In other words, .
Congruent Chords Have Congruent Minor Arcs
In the same circle or congruent circles, congruent chords have congruent minor arcs.
Proof. Draw the following diagram, in which the chords and are congruent.
Construct and by drawing the radii for the center to points and respectively.
Then, by the postulate.
This means that central angles,, which leads to the conclusion that .
Congruent Minor Arcs Have Congruent Chords and Congruent Central Angles
In the same circle or congruent circles, congruent chords have congruent minor arcs.
Proof. Draw the following diagram, in which the . In the diagram , and are radii of the circle.
Since , this means that the corresponding central angles are also congruent: .
Therefore, by the postulate.
We conclude that .
Here are some examples in which we apply the concepts and theorems we discussed in this section.
Example 1
Find the measure of each arc.
A.
B.
C.
A.
B.
C.
Example 2
Find in circle . The measures of all three arcs must add to .
Example 3
The circle goes through and . Find .
Draw the radii to points and .
We know that the measure of the minor arc is equal to the measure of the central angle.
Lesson Summary
In this section we learned about arcs and chords, and some relationships between them. We found out that there are major and minor arcs. We also learned that if two chords are congruent, so are the arcs they intersect, and vice versa.
Review Questions
In the circle identify the following:
four radii
a diameter
two semicircles
three minor arcs
two major arcs
Find the measure of each angle in :
Find the measure of each angle in :
The students in a geometry class were asked what their favorite pie is. The table below shows the result of the survey. Make a pie chart of this information, showing the measure of the central angle for each slice of the pie.
Kind of pie Number of students
apple
pumpkin
cherry
lemon
chicken
banana
total
Three identical pipes of diameter are tied together by a metal band as shown. Find the l
ength of the band surrounding the three pipes.
Four identical pipes of diameter are tied together by a metal band as shown. Find the length of the band surrounding the four pipes.
Review Answers
Some possibilities: .
Some possibilities: .
.
.
Chords
Learning Objectives
Find the lengths of chords in a circle.
CK-12 Geometry Page 34