9.1: Parts of Circles & Tangent Lines
Learning Objectives
 Define circle, center, radius, diameter, chord, tangent, and secant of a circle.
 Explore the properties of tangent lines and circles.
Review Queue
 Find the equation of the line with
m=−2 and passes through (4, 5).  Find the equation of the line that passes though (6, 2) and (3, 1).
 Find the equation of the line perpendicular to the line in #2 and passes through (8, 11).
Know What? The clock to the right is an ancient astronomical clock in Prague. It has a large background circle that tells the local time and the “ancient time” and then the smaller circle rotates around on the orange line to show the current astrological sign. The yellow point is the center of the larger clock. How does the orange line relate to the small and larger circle? How does the hand with the moon on it (black hand with the circle) relate to both circles? Are the circles concentric or tangent?
For more information on this clock, see: http://en.wikipedia.org/wiki/Prague_Astronomical_Clock
Defining Terms
Circle: The set of all points that are the same distance away from a specific point, called the center.
Radius: The distance from the center to the circle.
The center is typically labeled with a capital letter because it is a point. If the center is
Chord: A line segment whose endpoints are on a circle.
Diameter: A chord that passes through the center of the circle.
Secant: A line that intersects a circle in two points.
Tangent: A line that intersects a circle in exactly one point.
Point of Tangency: The point where the tangent line touches the circle.
Notice that the tangent ray
Example 1: Identify the parts of
a) A radius
b) A chord
c) A tangent line
d) The point of tangency
e) A diameter
f) A secant
Solution:
a)
b)
c)
d) Point
e)
f)
Coplanar Circles
Two circles can intersect in two points, one point, or no points. If two circles intersect in one point, they are called tangent circles.
Congruent Circles: Two circles with the same radius, but different centers.
Concentric Circles: When two circles have the same center, but different radii.
If two circles have different radii, they are similar. All circles are similar.
Example 2: Determine if any of the following circles are congruent.
Solution: From each center, count the units to the circle. It is easiest to count vertically or horizontally. Doing this, we have:
From these measurements, we see that
Notice that two circles are congruent, just like two triangles or quadrilaterals. Only the lengths of the radii are equal.
Tangent Lines
We just learned that two circles can be tangent to each other. Two triangles can be tangent in two different ways, either internally tangent or externally tangent.
If the circles are not tangent, they can share a tangent line, called a common tangent. Common tangents can be internally tangent and externally tangent too.
Notice that the common internal tangent passes through the space between the two circles. Common external tangents stay on the top or bottom of both circles.
Tangents and Radii
The tangent line and the radius drawn to the point of tangency have a unique relationship. Let’s investigate it here.
Investigation 91: Tangent Line and Radius Property
Tools needed: compass, ruler, pencil, paper, protractor
 Using your compass, draw a circle. Locate the center and draw a radius. Label the radius
AB¯¯¯¯¯¯¯¯ , withA as the center.  Draw a tangent line,
BC←→ , whereB is the point of tangency. To draw a tangent line, take your ruler and line it up with pointB . Make sure thatB is the only point on the circle that the line passes through.  Using your protractor, find
m∠ABC .
Tangent to a Circle Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.
To prove this theorem, the easiest way to do so is indirectly (proof by contradiction). Also, notice that this theorem uses the words “if and only if,” making it a biconditional statement. Therefore, the converse of this theorem is also true.
Example 3: In
Solution: Because
Example 4: Find
Solution:
Example 5: Determine if the triangle below is a right triangle. Explain why or why not.
Solution: To determine if the triangle is a right triangle, use the Pythagorean Theorem.
Example 6: Find the distance between the centers of the two circles. Reduce all radicals.
Solution: The distance between the two circles is
Tangent Segments
Let’s look at two tangent segments, drawn from the same external point. If we were to measure these two segments, we would find that they are equal.
Theorem 102: If two tangent segments are drawn from the same external point, then the segments are equal.
The proof of Theorem 102 is in the review exercises.
Example 7: Find the perimeter of
Solution:
We say that \begin{align*}\bigodot G\end{align*} is inscribed in \begin{align*}\triangle ABC\end{align*}. A circle is inscribed in a polygon, if every side of the polygon is tangent to the circle.
Example 8: If \begin{align*}D\end{align*} and \begin{align*}A\end{align*} are the centers and \begin{align*}AE\end{align*} is tangent to both circles, find \begin{align*}DC\end{align*}.
Solution: Because \begin{align*}AE\end{align*} is tangent to both circles, it is perpendicular to both radii and \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DBE\end{align*} are similar. To find \begin{align*}DB\end{align*}, use the Pythagorean Theorem.
\begin{align*}10^2+24^2 &= DB^2\\ 100+576 &= 676\\ DB &= \sqrt{676}=26\end{align*}
To find \begin{align*}BC\end{align*}, use similar triangles.
\begin{align*}\frac{5}{10}=\frac{BC}{26} & \longrightarrow BC=13\\ DC=AB+BC &= 26+13=39\end{align*}
Example 9: Algebra Connection Find the value of \begin{align*}x\end{align*}.
Solution: Because \begin{align*}\overline{AB} \bot \overline{AD}\end{align*} and \begin{align*}\overline{DC} \bot \overline{CB}, \overline{AB}\end{align*} and \begin{align*}\overline{CB}\end{align*} are tangent to the circle and also congruent. Set \begin{align*}AB = CB\end{align*} and solve for \begin{align*}x\end{align*}.
\begin{align*}4x9 &= 15\\ 4x &= 24\\ x &= 6\end{align*}
Know What? Revisited Refer to the photograph in the “Know What?” section at the beginning of this chapter. The orange line (which is normally black, but outlined for the purpose of this exercise) is a diameter of the smaller circle. Since this line passes through the center of the larger circle (yellow point, also outlined), it is part of one of its diameters. The “moon” hand is a diameter of the larger circle, but a secant of the smaller circle. The circles are not concentric because they do not have the same center and are not tangent because the sides of the circles do not touch.
Review Questions
Determine which term best describes each of the following parts of \begin{align*}\bigodot P\end{align*}.
 \begin{align*}\overline{KG}\end{align*}
 \begin{align*}\overleftrightarrow{FH}\end{align*}
 \begin{align*}\overline{KH}\end{align*}
 \begin{align*}E\end{align*}
 \begin{align*}\overleftrightarrow{BK}\end{align*}
 \begin{align*}\overleftrightarrow{CF}\end{align*}
 \begin{align*}A\end{align*}
 \begin{align*}\overline{JG}\end{align*}
 What is the longest chord in any circle?
Copy each pair of circles. Draw in all common tangents.
Coordinate Geometry Use the graph below to answer the following questions.
 Find the radius of each circle.
 Are any circles congruent? How do you know?
 Find all the common tangents for \begin{align*}\bigodot B\end{align*} and \begin{align*}\bigodot C\end{align*}.
 \begin{align*}\bigodot C\end{align*} and \begin{align*}\bigodot E\end{align*} are externally tangent. What is \begin{align*}CE\end{align*}?
 Find the equation of \begin{align*}\overline{CE}\end{align*}.
Determine whether the given segment is tangent to \begin{align*}\bigodot K\end{align*}.
Algebra Connection Find the value of the indicated length(s) in \begin{align*}\bigodot C\end{align*}. \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are points of tangency. Simplify all radicals.

\begin{align*}A\end{align*} and \begin{align*}B\end{align*} are points of tangency for \begin{align*}\bigodot C\end{align*} and \begin{align*}\bigodot D\end{align*}, respectively.
 Is \begin{align*}\triangle AEC \sim \triangle BED\end{align*}? Why?
 Find \begin{align*}BC\end{align*}.
 Find \begin{align*}AD\end{align*}.
 Using the trigonometric ratios, find \begin{align*}m \angle C\end{align*}. Round to the nearest tenth of a degree.
 Fill in the blanks in the proof of Theorem 102. Given: \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CB}\end{align*} with points of tangency at \begin{align*}A\end{align*} and \begin{align*}C\end{align*}. \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{DC}\end{align*} are radii. Prove: \begin{align*}\overline{AB} \cong \overline{CB}\end{align*}
Statement  Reason 

1.  
2. \begin{align*}\overline{AD} \cong \overline{DC}\end{align*}  
3. \begin{align*}\overline{DA} \bot \overline{AB}\end{align*} and \begin{align*}\overline{DC} \bot \overline{CB}\end{align*}  
4.  Definition of perpendicular lines 
5.  Connecting two existing points 
6. \begin{align*}\triangle ADB\end{align*} and \begin{align*}\triangle DCB\end{align*} are right triangles  
7. \begin{align*}\overline{DB} \cong \overline{DB}\end{align*}  
8. \begin{align*}\triangle ABD \cong \triangle CBD\end{align*}  
9. \begin{align*}\overline{AB} \cong \overline{CB}\end{align*} 
 From the above proof, we can also conclude (fill in the blanks):
 \begin{align*}ABCD\end{align*} is a _____________ (type of quadrilateral).
 The line that connects the ___________ and the external point \begin{align*}B\end{align*} _________ \begin{align*}\angle ADC\end{align*} and \begin{align*}\angle ABC\end{align*}.
 Points \begin{align*}A, B, C\end{align*}, and \begin{align*}D\end{align*} are all points of tangency for the three tangent circles. Explain why \begin{align*}\overline{AT} \cong \overline{BT} \cong \overline{CT} \cong \overline{DT}\end{align*}.
 Circles tangent at \begin{align*}T\end{align*} are centered at \begin{align*}M\end{align*} and \begin{align*}N\end{align*}. \begin{align*}\overline{ST}\end{align*} is tangent to both circles at \begin{align*}T\end{align*}. Find the radius of the smaller circle if \begin{align*}\overline{SN} \bot \overline{SM}\end{align*}, \begin{align*}SM=22, TN=25\end{align*} and \begin{align*}m \angle SNT=40^\circ\end{align*}.
 Four circles are arranged inside an equilateral triangle as shown. If the triangle has sides equal to 16 cm, what is the radius of the bigger circle? What are the radii of the smaller circles?
 Circles centered at \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are tangent at \begin{align*}W\end{align*}. Explain why \begin{align*}A, B\end{align*} and \begin{align*}W\end{align*} are collinear. \begin{align*}\overline{TU}\end{align*} is a common external tangent to the two circles. \begin{align*}\overline{VW}\end{align*} is tangent to both circles. Justify the fact that \begin{align*}\overline{TV} \cong \overline{VU} \cong \overline{VW}\end{align*}.
Review Queue Answers
 \begin{align*}y = 2x+3\end{align*}
 \begin{align*}y = \frac{1}{3} x\end{align*}
 \begin{align*}y = 3x  13\end{align*}
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