10.3 Lesson: Circles and Tangents

10.3 Lesson: Circles and Tangents

Circles and Tangents

• I'm sorry that I cannot be with you for today's lesson.

• A video will be added to Canvas at a later date with me going through these notes.

• This activity will be taken as a grade!

Example 1:

Q is the center of this triangle.

a) Name the circle:

b) Name a radius shown:

c) What is the length of the radius?

d) What is the length of the diameter?

e) Name all interior points shown:

f) Name all exterior points shown:

Example 1: Answers

Q is the center of this triangle.
a) Circle Q
b)  $\overline{QP},\ \overline{QT}$  
c) r = 16 m
d) d = 32 m
e) Interior Points: Points R, Q, and S
f) Exterior Points: X

Multiple Choice

A chord is a line segment with ________ on the circle.

• endpoints

• multiple notes

Multiple Choice

A secant is a line that intersects 2 points and _______ into the exterior of the circle.

• extends

• wiggles

Multiple Choice

A tangent is a line, line segment, or ray that intersects a circle at one _______.

• point

• plane

Example 2:

Name Examples of each.

a) Center:

b) All Radii:

c) All Chords:

d) All Secants:

e) Diameter:

f) Tangent:

g) Point of Tangency:

h) Interior Points:

i) Exterior Points

Example 2: Answers

a) Point O
b)  $\overline{ON},\ \overline{OR},\ and\ \overline{OM}$  
c)  $\overline{MN}\ and\ \overline{MT}$  
d)  $\overleftarrow{M}\overrightarrow{T}$  
e)  $\overline{MN}$  
f)  $\overleftarrow{P}\overrightarrow{S}$  
g) Point Q
h) Interior Points: B, I and O
i) Exterior Points: P and S

Tangents Line to Circle Theorem

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

Multiple Choice

 By the Tangent Line to Circle Theorem, $m\angle CBA\ =\ 90\degree$ . Use Pythagorean's Theorem to find the value of CA.

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Multiple Choice

Now that you know the length of CA, what is the length of x in this picture/

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External tangent Congruence Theorem

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Multiple Choice

(Example 4) By the External Tangent Congruence Theorem, QP = RP. Use this information to find the value of x.

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External Tangent Congruence Theorem

• When circles are inscribed in polygons, the polygons are said to be circumscribed polygons. For these polygons, each side is tangent to the circle!

• Recall that inscribed means drawn inside and circumscribed means drawn around.

Example 5

 $\Delta TRW$  is circumscribed about Circle A. The perimeter of  $\Delta TRW$  is 50, TK = 3, and WM = 9.5. our goal is to find the length of TR.

Example 5

By the External Tangent Congruence Theorem, since TK = 3, we know that TL = 3 also.

 $\overline{TK}$  and  $\overline{TL}$  are both tangent to Circle A from the same point T, so TL = TK.

Example 5

Likewise, WK = WM and RL = RM. 

The next slide has to solution to Example 5.

Multiple Choice

(Example 6) in the figure $\overleftarrow{R}\overrightarrow{P}$ 

is tangent to circle Q at point R. Find the radius of circle Q.

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Multiple Choice

(Example 7) Find the value of x.

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Multiple Choice

(Example 7) Find the value of $m\angle ABC.$ 

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Multiple Choice

(Example 7) Find the value of BC.

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Multiple Choice

(Example 7) Find the diameter of circle C.

• 20 $\pi$  

• 40 $\pi$  

• 10 $\pi$  

• 400 $\pi$  

Multiple Choice

(Example 8) Find the perimeter of the polygon that circumscribes the circle.

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