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Circles With and Without Coordinates

Tangent Line to a Circle

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 O and point P 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 PO
Using a straightedge, draw the segment connecting the point P and the center of the circle O.

2

Mark Midpoint M of PO
To locate the midpoint of PO, the perpendicular bisector of PO can be used. Recall that there are three steps to draw a perpendicular bisector.
The point where PO and its perpendicular bisector intersect is the midpoint of PO.

3

Draw a Circle with Center M and Radius MO
Next, place the compass tip on M and stretch the compass to O, or P. Then, draw the circle M.

4

Draw Tangents
The points where the circles intersect will be the points of tangency for the tangents drawn from P to the circle O.

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

Why

Consider OA, a radius of

In circle M, OAP is an inscribed angle on a diameter of Since an inscribed angle opposite the diameter is a right angle, OAP 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, PA 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 AB and circle O. The segment's endpoint A can be moved anywhere outside of While endpoint B can be moved anywhere. An eye-like shape appears on the screen when AB is tangent to the circle. Give it a try!

Kriz can't quite place point B 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 A is a point outside B should be the point of tangency in order for AB to be tangent to the circle. On the example shape, by extending AB, it can be observed that B is the point of tangency.

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

  1. Connect the outer point and the center of the circle.
  2. Find midpoint M of the segment drawn.
  3. Draw a circle with center M and radius half the segment drawn.
  4. Draw tangent.

In this case, point A 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 B 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 A. When A is reflected across the line that passes through P and O, 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 AB and AC are tangent segments to then ABAC.

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 B and C are points of tangency. Therefore, by the Tangent to Circle Theorem, B and C are right angles. Consequently, ABO and ACO are right triangles.

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

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

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

ABAC


Example

External Tangent Congruence Theorem

In the diagram, all three segments are tangent to circle P.

The points D, E, and F are the points where the segments touch the circle. If AB=12, BC=10, and CA=6, find AE.

Solution

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

Smilarly, BE and BF, and CF and CD 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 AE, the Elimination Method can be used. Start by multiplying the second equation by -1.
Adding this equation to the first equation will eliminate BE.
Simplify
AE+BEBECF=1210
AECF=2
Since -CF and CF are additive inverses, adding this equation to the third equation will eliminate CF, and thus give AE.
CF+AE+(AECF)=6+(2)
Simplify
CF+AE+AECF=6+2
2AE=8
AE=4

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 30, 1986, in a packed stadium full of fans, Yuriy Sedykh set the world record with a throw of 86.74 meters. Perhaps a throw farther than the superhero!
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