Understanding Circles: Tangents, Inscribed Angles, and Theorems

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Chapter 11 Circles
 
11-1 Tangent Lines
 
A 
tangent to a circle
 is a line in the
plane of the circle that intersects
the  circle in exactly ONE point.
The point where a circle and a
tangent intersect is the 
point of
tangency.
 
Theorem 11-1
 
If a line is tangent to a circle,
then the line is perpendicular to
the radius drawn to the point of
tangency.
 
Example: ML and MN are tangent to
Circle O. Find the value of x.
 
Example: ED is tangent to Circle O. Find
the value of x.
 
Theorem 11-2
 
If a line in the plane of a circle is
perpendicular to a radius at its
endpoint on the circle, then the
line is tangent to the circle.
 
Example: Is ML tangent to Circle N at
Point L? Explain
 
Theorem 11-3
 
The two segments tangent to a
circle from a point outside the
circle are congruent.
 
Example: Circle O is inscribed in Triangle
ABC. Find the perimeter of Triangle ABC.
 
Example: Circle O is inscribed in
Triangle PQR. Triangle PQR has a
perimeter of 46. Find QY.
 
12
 
6
 
11- 3 Inscribed Angles
 
Inscribed angle
 – An angle is
inscribed in a circle if the vertex of
the angle is on the circle and the
sides of the angle are chords of the
circle.
Intercepted arc 
– An intercepted
arc is an arc of a circle having
endpoints on the sides of an
inscribed angle, and its other
points in the interior of the angle.
 
Theorem 11-9 Inscribed Angle Theorem
 
The measure of an inscribed angle
is half the measure of its
intercepted arc.
 
Example: Find the values of a and b.
 
Examples
 
11-4 Angle Measures and Segment Lengths
 
A 
secant
 is a line that intersects a
circle at TWO points.
 
Theorem 11-11
 
The measure of an angle formed by
two lines that intersect 
INSIDE
 a
circle is half the 
sum
 of the
measures of the intercepted arcs.
 
Formula:
 
Theorem 11-11
 
The measure of an angle formed by
two lines that intersect 
OUTSIDE
 a
circle is half the 
difference
 of the
measures of the intercepted arcs.
 
Formula:
 
Examples:
Find the value of each variable
 
Examples:
Find the value of each variable
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Explore the concept of circles in geometry, focusing on tangent lines, inscribed angles, and related theorems. Understand the properties of tangents, relationships between angles and arcs, and how to apply theorems in circle problems. Visual examples and explanations included.


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  1. Chapter 11 Circles

  2. 11-1 Tangent Lines A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly ONE point. The point where a circle and a tangent intersect is the point of tangency.

  3. Theorem 11-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

  4. Example: ML and MN are tangent to Circle O. Find the value of x.

  5. Example: ED is tangent to Circle O. Find the value of x.

  6. Theorem 11-2 If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

  7. Example: Is ML tangent to Circle N at Point L? Explain

  8. Theorem 11-3 The two segments tangent to a circle from a point outside the circle are congruent.

  9. Example: Circle O is inscribed in Triangle ABC. Find the perimeter of Triangle ABC.

  10. Example: Circle O is inscribed in Triangle PQR. Triangle PQR has a perimeter of 46. Find QY. 12 6

  11. 11- 3 Inscribed Angles Inscribed angle An angle is inscribed in a circle if the vertex of the angle is on the circle and the sides of the angle are chords of the circle. Intercepted arc An intercepted arc is an arc of a circle having endpoints on the sides of an inscribed angle, and its other points in the interior of the angle.

  12. Theorem 11-9 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc.

  13. Example: Find the values of a and b.

  14. Examples

  15. 11-4 Angle Measures and Segment Lengths A secant is a line that intersects a circle at TWO points.

  16. Theorem 11-11 The measure of an angle formed by two lines that intersect INSIDE a circle is half the sum of the measures of the intercepted arcs. Formula:

  17. Theorem 11-11 The measure of an angle formed by two lines that intersect OUTSIDE a circle is half the difference of the measures of the intercepted arcs. Formula:

  18. Examples: Find the value of each variable

  19. Examples: Find the value of each variable

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