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5. Tangent Line to a Circle
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Chapter 5
5. 

Tangent Line to a Circle

Tangent lines to circles have been a topic of interest and study in geometry. This lesson delves into the methods of constructing a tangent line to a circle from an external point. It emphasizes the properties of tangents and how to ascertain that a drawn line is indeed a tangent. The lesson further elaborates on drawing a tangent line through an outer point using a compass and straightedge. The process involves connecting the external point to the circle's center, marking the midpoint, drawing a circle with the midpoint as the center, and finally drawing the tangents. The lesson also touches upon the real-life application of tangents, illustrating it with an example of a hammer throw in the Olympics.
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Lesson Settings & Tools
8 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Tangent Line to a Circle
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Tangent lines have been defined and studied earlier in this course. In this lesson, methods for constructing a tangent line to a circle from an external point will be discussed.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Challenge

Investigating Properties of Tangents to a Circle

Consider a circle with center and point outside of the circle. Using a straightedge and compass, can you construct a tangent to the circle through the given point?

Circle and External Point
Think about the properties of tangents. How can you justify that the line you draw is indeed the tangent?
Discussion

Drawing a Tangent Line to a Circle Through an Outer Point

Given a circle and a point outside the circle, a compass and straightedge can be used to draw a tangent line from the point to the circle.

There are four steps to construct a tangent line.
1
Draw
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Using a straightedge, draw the segment connecting the point and the center of the circle
2
Mark Midpoint of
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To locate the midpoint of the perpendicular bisector of can be used. Recall that there are three steps to draw a perpendicular bisector.
The point where and its perpendicular bisector intersect is the midpoint of
3
Draw a Circle with Center and Radius
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Next, place the compass tip on and stretch the compass to or Then, draw the circle
4
Draw Tangents
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The points where the circles intersect will be the points of tangency for the tangents drawn from to the circle

The Inscribed Right Triangle Theorem can be used to justify why this construction works.

Why

Consider a radius of

In circle is an inscribed angle on a diameter of Since an inscribed angle opposite the diameter is a right angle, is a right angle.

As can bee seen, The radius of is perpendicular to a line that passes through a point on the circle. Therefore, by the Tangent to Circle Theorem, is tangent to the circle.

Example

Using Tangents to a Circle to Construct an Eye

Kriz is learning a graphic program. By default, the program shows segment and circle The segment's endpoint can be moved anywhere outside of While endpoint can be moved anywhere. An eye-like shape appears on the screen when is tangent to the circle. Give it a try!
Kriz cannot quite place point in position to see the eye-like shape appear. Help Kriz out!

Answer

See solution.

Hint

How can a tangent line from a point outside of the given circle be constructed?

Solution

Since point is a point outside should be the point of tangency in order for to be tangent to the circle. On the example shape, by extending it can be observed that is the point of tangency.

Constructing a tangent from an outer point will help locate the point of tangency for a tangent drawn from Recall the steps in constructing a tangent.

  1. Connect the outer point and the center of the circle.
  2. Find midpoint of the segment drawn.
  3. Draw a circle with center and radius half the segment drawn.
  4. Draw tangent.
In this case, point is the outer point through which the tangent line is drawn. To get the example shape, move point A to the left as shown and then follow the steps.
As can be seen, the points where the circles intersect are the points of tangency. Therefore, point should be on these points.
Discussion

Relationship Between Tangents and Lines of Symmetry

The lines of symmetry of a circle are the lines that passes through the center of the circle.

Suppose that a tangent line drawn from an outer point intersects a circle at When is reflected across the line that passes through and its image will also be a point of tangency for another tangent.

Discussion

External Tangent Congruence Theorem

Two tangent segments drawn from a common external point to the same circle are congruent.

For the above diagram, the following conditional statement holds true.

If and are tangent segments to then

Proof


Consider two triangles.

  • The triangle formed by the radius the segment and the tangent segment
  • The triangle formed by the radius the segment and the tangent segment

These two triangles can be visualized in the diagram.

Note that and are points of tangency. Therefore, by the Tangent to Circle Theorem, and are right angles. Consequently, and are right triangles.

Because all radii of the same circle are congruent, it can be said that and are congruent. Moreover, and share the same hypotenuse By the Reflexive Property of Congruence, is congruent to itself.

Combining all of this information, it can be said that the hypotenuse and one leg of are congruent to the hypotenuse and the corresponding leg of

Therefore, by the Hypotenuse-Leg Theorem, and are congruent triangles. Since corresponding parts of congruent figures are congruent, it can be said that and are congruent.


Example

External Tangent Congruence Theorem

In the diagram, all three segments are tangent to circle

The points and are the points where the segments touch the circle. If and find

Hint

Use the External Tangent Congruence Theorem.

Solution

From the graph, it can be seen that and are tangent segments with a common endpoint outside By the External Tangent Congruence Theorem, and are congruent.

Smilarly, and and and are congruent tangent segments.
Using the Segment Addition Postulate and the given lengths, a system of equations with three equations and three unknowns can be written.
To find the Elimination Method can be used. Start by multiplying the second equation by
Adding this equation to the first equation will eliminate
▼
Simplify
AddNeg

SubTerm

Subtract term

Since and are additive inverses, adding this equation to the third equation will eliminate and thus give
▼
Simplify
RemovePar

Remove parentheses

AddSubFrac

Add and subtract fractions

DivEqn
Closure

Tangents to Circles in Real Life

Imagine a superhero joining the Olympics to throw a hammer. An athlete would typically spin counterclockwise three or four (rarely five) times, then release the hammer. As viewed from above, the hammer travels on a path that is tangent to the circle created when the athlete spins. The diagram below shows the path of the superhero's hammer throw. See how the super hero fairs! Note, it is a not-to-scale drawing.
On August in a packed stadium full of fans, Yuriy Sedykh set the world record with a throw of meters. Perhaps a throw farther than the superhero!



Level 1
Level 2
Level 3
Tangent Line to a Circle
Exercises

Is tangent to the circle?

triangle ABC and circle A
triangle ABC and a circle A
triangle ABC with its small leg being the diameter of circle A

Suppose a segment is perpendicular to a radius at its endpoint on a circle's circumference. In that case, we know by the Tangent to Circle Theorem that this segment is tangent to the circle. Therefore, to determine if BC is tangent to the circle, we can use the Pythagorean Theorem to evaluate if △ ABC is a right triangle.

Since we obtained a true statement, the triangle is, in fact, a right triangle. This means BC is tangent to the circle ⊙ A.


Notice that we only know part of the length of AC. However, the unknown part is also the circle's radius, which means AC must be the sum of the known part and the circle's radius.

Now we can use the Pythagorean Theorem to determine if △ ABC is a right triangle.

As we can see, △ ABC is a right triangle. Therefore, BC is tangent to the circle.


As in previous parts, we will use the Pythagorean Theorem to evaluate if BC is tangent to the circle. Notice that one of the triangle's legs is the diameter of the circle. Since we have been given the radius, we find the diameter by multiplying this value by 2.

Let's substitute the legs and hypotenuse in the Pythagorean Theorem and determine if it makes a true statement or not.

Since the Pythagorean Theorem is not satisfied, △ ABC is not a right triangle. Therefore, BC can not be tangent to the circle.

is tangent to Find the radius of Answer in exact form.

A circle with a right triangle ABC where B is on the circles circumference
A circle and triangle ABC where B is tangent to the circle

Notice that △ ABC is a right triangle. This triangle has legs of r and 24 units and a hypotenuse of (r+18) units. Let's mark these lengths in the diagram.

If we substitute the hypotenuse and the legs into the Pythagorean Theorem, we can solve for r.

We found that the radius r is 7.

As in Part A, we have a right triangle. Let's identify its legs and hypotenuse.

As in Part A, we will substitute the length of the triangle's legs and hypotenuse into the Pythagorean Theorem to determine the radius r.

The radius r is 163.

Determine the value of if we know that and are tangent segments to the circle.

circle with two tangents to the circle which coincides at the common exterior point of A
circle with tangents to the circle which intersect at A

According to the External Tangent Congruence Theorem, if two line segments drawn from the same exterior point are tangent to a circle, then they are congruent.

With this information, we can write an equation that we can solve for x.


As in Part A, we will equate the expressions for the two tangents and solve for x.


What is the perimeter of if is inscribed in the quadrilateral.

Circle Y inscribed in the quadrilateral TVWX

According to the External Tangent Congruence Theorem, tangents drawn from a common external point are congruent segments. This means XD and XA are congruent, as are UD and UC. Let's add this information to the diagram.

Now we can calculate XD, and thereby also XA, by subtracting UD from XU.

Let's add the length of XD and XA to the diagram.

Now we can determine the perimeter of TVUX by adding all of the sides.

The following circle represents a cross section of Earth where is an arbitrary height above the ground and is the line of sight across the horizon. The radius of the Earth is about km.

person standing h km above the Earth watching d km across the horizon
 What is if kilometers? Round to the nearest integer.
What is if kilometers? Round to one decimal place.

We know that the radius of Earth is about 6400 kilometers. We want to know the distance d one can see over the horizon when we are 5 kilometers above the Earth.

We see that ET= 6400 and EH= 6400+ 5. To find d, we will use the Pythagorean Theorem.

The distance a person can see when standing 5 km above the Earth is about 253 km.


Now that we know the distance one sees over the horizon. Let's update our illustration.

By substituting the legs and hypotenuse into the Pythagorean Theorem, we can determine h.

To solve this equation, we will use the Quadratic Formula.

When we see 500 kilometers across the horizon, we are about 19.5 kilometers above the ground!

Tangent Line to a Circle

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