Lesson 5: Tangent Lines and the Tangent Function

NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 5 PRECALCULUS AND ADVANCED TOPICS
Lesson 5: Tangent Lines and the Tangent Function
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Student Outcomes
Students construct a tangent line from a point outside a given circle to the circle (G-C.A.4).
Lesson Notes
This lesson is designed to address standard G-C.A.4, which involves constructing tangent lines to a given circle from a
point outside the circle. The lesson begins by revisiting the geometric origins of the tangent function from Algebra II
Module 2 Lesson 6. Students then explore the standard by means of construction by compass and straightedge and by
paper folding. Students are provided several opportunities to create mathematical arguments to explain their
constructions.
Classwork
Exercises 1–4 (4 minutes)
In this sequence of exercises, students recall the connections between the trigonometric functions and the geometry of
the unit circle.
Exercises 1–12
The circle shown to the right is a unit circle, and the length of is

radians.
)?
The sine of
is the vertical component of point , so has
length (
).
)?
is the horizontal component of point , so
has length (
).
)?
=

=
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)?
=

Discussion (7 minutes): Connections Between Geometry and Trigonometry
Do you recall why the length of is called the tangent of
3 ? And do you recall why the length of is called
the secant of
3 ? (Hint: Look at how lines containing and are related to the circle.)
The line containing is tangent to the circle because it intersects the circle once, so it makes sense to
refer to as a tangent segment and to refer to the length as the tangent of
3 .
The line containing is a secant line because it intersects the circle twice, so it makes sense to refer to
as a secant segment and to refer to the length as the secant of
3 .
How can you be sure that the line containing is, in fact, a tangent line?
We see that is perpendicular to segment at point , and so must be tangent to the circle.
Explain how this diagram can be used to show that tan (
3 ) =
.
The triangles in the diagram have two pairs of congruent angles, so they are similar to each other. It
follows that their sides are proportional, and so tan(

3 )
3 ),
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Explain how this diagram can be used to show that sin2 ( 3 ) + cos2 (
3 ) = 1.
If we apply the Pythagorean theorem to , we find that ()2 + ()2 = ()2, so sin2 ( 3 ) +
cos2 ( 3 ) = 1.
Which trigonometric identity can be established by applying the Pythagorean theorem to ?
tan2 ( 3 ) + 12 = sec2 (
3 )
What is the scale factor that relates the sides of to those of ?
Point is the midpoint of , so the sides of are twice the length of the corresponding sides of
.
Explain how the diagram above can be used to show that tan ( 3 ) = √3.
Since the length of is
3 radians, it follows that is equilateral, and so its sides must each be 2
units long. Applying the Pythagorean theorem to , we find 12 + 2 = 22, which means that
2 = 3, and so = √3. Thus, tan ( 3 ) = √3.
This diagram contains a host of information about trigonometry. But can we take this diagram even further?
Let’s press on to some new territory. Much of the remaining discussion is devoted to the topic of
constructions. Get ready to have some fun with paper folding and your compass and straightedge!
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Exploratory Challenge 1 (7 minutes): Constructing Tangents via Paper-Folding
Earlier we observed that is tangent to the circle. Can you visualize another line through point that is also
tangent to the circle? Try to draw a second tangent line through point , and label its point of intersection
with the circle as point .
First, let’s examine this diagram through the lens of transformations. Can you see how to map onto ?
It appears that you could map onto by reflecting it across .
Let’s get some hands-on experience with this by performing the reflection in the most natural way
imaginable—by folding a piece of paper.
Give students an unlined piece of copy paper or patty paper, a ruler, and a compass.
The goal of this next activity is essentially to recreate the diagram above. Students take one tangent line and use a
reflection to produce a second tangent line.
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Choose a point in the middle of your paper, and label it . Use your compass to draw a circle with center .
Choose a point on that circle, and label it . Use your straightedge to draw the line through and ,
extending it beyond the circle. Can you see how to fold the paper in such a way that the crease is tangent to
the circle at point ? Think about this for a moment.
We want to create a line that is perpendicular to at point , so we fold the paper in such a way that
the crease is on point and maps to itself.
Now choose a point on the crease, and label it . Try to create a second tangent line through point . Can you
see how to do this?
We just need to fold the paper so that the crease is on points and . We can see through the back of
the paper where point is, and this is where we mark a new point , which is the other point of
tangency. In other words, is also tangent to the circle.
Do you think you could construct an argument that is truly tangent to the circle? Think about this for a few
minutes, and then share your thoughts with the students around you.
A reflection is a rigid motion, preserving both distances and angles. Since is tangent to the circle,
we know that ∠ is a right angle. And since the distance from to is preserved by the reflection,
it follows that point , which is the image of , is also located on the circle because = , both of
which are radii of the circle. This means that is on the circle and that ∠ is a right angle, which
proves that is indeed tangent to the circle.
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Exploratory Challenge 2 (5 minutes): Constructing Tangents Using a Compass
Let’s try to produce this tangent line without using paper-folding. Can you see how to produce this diagram
using only a compass and a straightedge? Take a few minutes to explore this problem.
We use a straightedge to draw a line that is perpendicular to at point and choose a point on
this perpendicular line as before. Next, we use a compass to draw a circle around point that contains
point . This circle will intersect the original circle at a point , which is the desired point of tangency.
Now make an argument that your construction produces a line that is truly tangent to the circle.
Each point on the circle around point is the same distance from , so = . Also, each point on
the circle around point is the same distance from point , so = . Since = ,
≅ by the SSS criterion for triangle congruence. Since ∠ is a right angle, it follows by
CPCTC that ∠ is also a right angle. This proves that is indeed tangent to the circle.
Exploratory Challenge 3 (7 minutes): Constructing the Tangents from an External Point
Now let’s change the problem slightly. Suppose we are given a circle and an external point. How might we go
about constructing the lines that are tangent to the given circle and pass through the given point? Take
several minutes to wrestle with this problem, and then share your thoughts with a neighbor.
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This is a tough problem, isn’t it? Let’s see if we can solve it by using some logical reasoning. What do you
know about the lines you want to create?
We want the lines to be tangent to the circle, so they must be constructed in such a way that ∠ and
∠ are right angles.
Perhaps we can use this to our advantage. Let’s consider the entire locus of points such that ∠ is a right
angle. Can you describe this locus? Can you construct it? This is a challenge in its own right. Let’s explore this
challenge using a piece of paper, which comes with some built-in right angles.
Take a blank piece of paper, and mark two points several inches apart; label them and . Now take a second
piece of paper, and line up one edge on and the adjacent edge on . Mark the point where the corner of the
paper is, then shift the corner to a new spot, keeping the edges against and . Quickly repeat this 20 or so
times until a clear pattern emerges. Then describe what you see.
This locus of points is a circle with diameter .
Now construct this circle using your compass.
We just need to construct the midpoint of , which is the center of the circle.
We are ready to apply this result to our problem involving tangents to a circle. Do you see how this relates to
the problem of creating a tangent line?
By drawing a circle with diameter , we create two points and with two important properties.
First, they are both on the circle around . Second, they are on the circle with diameter , which
means that ∠ and ∠ are right angles, just as we require for tangent lines.
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 5 PRECALCULUS AND ADVANCED TOPICS
84
Instruct students to perform the following exercises and to compare their results with a partner.
5. Use a compass to construct the tangent lines to the given circle that pass through the given point.
Sample solution:
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6. Analyze the construction shown below. Argue that the lines shown are tangent to the circle with center .
Since point is on circle , we know that ∠ is a right angle. Since point is on circle and ∠ is a right
angle, it follows that is tangent to circle . In the same way, we can show that is tangent to circle .
7. Use a compass to construct a line that is tangent to the circle below at point . Then choose a point on the
tangent line, and construct another tangent to the circle through .
Sample solution:
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Instruct students to perform the following exercises and to compare their results with a partner.
8. The circles shown below are unit circles, and the length of is
radians.
.
9. Which trigonometric function corresponds to the length of ?
The length of represents the cosecant of
.
10. Which trigonometric identity gives the relationship between the lengths of the sides of ?
(
We have
)? What is the value of (
)? Use the Pythagorean theorem to support your
answers.
Let = . Then we have + = () = , which means = ; therefore, = √
. So (
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Closing (2 minutes)
If you are given a circle and an external point, how do you use a compass to construct the lines that pass
through the given point and are tangent to the given circle? Use your notebook to write a summary of the
main steps involved in this construction.
Let’s say the center of the circle is point and the external point is . First, we need to construct the
midpoint of the segment joining and —call this point . Next, we draw a circle around point that
goes through and . The points where this circle intersects the first circle are the points of tangency,
so we just connect these points to point to form the tangents.
Exit Ticket (4 minutes)
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Name Date
Exit Ticket
1. Use a compass and a straightedge to construct the tangent lines to the given circle that pass through the given
point.
2. Explain why your construction produces lines that are indeed tangent to the given circle.
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Exit Ticket Sample Solutions
1. Use a compass and a straightedge to construct the tangent lines to the given circle that pass through the given point.
2. Explain why your construction produces lines that are indeed tangent to the given circle.
Since points and are on circle , ∠ and ∠ are right angles. Thus, and are tangent to circle .
Problem Set Sample Solutions
1. Prove Thales’ theorem: If , , and are points on a circle where is a diameter of the circle, then ∠ is a
right angle.
Since = = , and are isosceles triangles. Therefore,
∠ = ∠, and ∠ = ∠.
Let ∠ = and ∠ = . The sum of three internal angles of
equals °.
Therefore, + ( + ) + = °, so + = °, and + = °. Since
∠ = + , we have ∠ = °, so ∠ is a right angle.
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2. Prove the converse of Thales’ theorem: If is a diameter of a circle and is a point so that ∠ is a right angle,
then lies on the circle for which is a diameter.
Construct the right triangle, .
Construct the line that is parallel to through point .
Construct the line that is parallel to through point .
Let be the intersection of lines and .
The quadrilateral forms a parallelogram by construction.
By the properties of parallelograms, the adjacent angles are supplementary. Since ∠ is a right angle, it follows
that angles ∠, ∠, and ∠ are also right angles. Therefore, the quadrilateral is a rectangle.
Let be the intersection of the diagonals and . Then, by the properties of parallelograms, point is the
midpoint of and , so = = = . Therefore, is the center of the circumscribing circle, and the
hypotenuse of , , is a diameter of the circle.
3. Construct the tangent lines from point to the circle given below.
Mark any three points , , and on the circle, and construct perpendicular bisectors of and .
Let be the intersection of the two perpendicular bisectors.
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Construct the midpoint of .
Construct a circle with center and radius .
The circle centered at will intersect the original circle at points and .
Construct two tangent lines and .
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4. Prove that if segments from a point are tangent to a circle at points and , then ≅ .
Let be a point outside of a circle with center , and let and be points on the circle so that and are
tangent to the circle. Then, = , = , and ∠ = ∠ = °, so ≅ by the
Hypotenuse Leg congruence criterion. Therefore, ≅ because corresponding parts of congruent triangles are
congruent.
5. Given points , , and so that = , construct a circle so that is tangent to the circle at and is
tangent to the circle at .
Construct a perpendicular bisector of .
Construct a perpendicular bisector of .
The perpendicular bisectors will intersect at point .
Construct a line through points and .
Construct a circle with center and radius .
The circle centered at will intersect at .
Construct a circle centered at with radius .

Lesson 5: Tangent Lines and the Tangent Function

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Lesson 5: Tangent Lines and the Tangent Function

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