2. • This Slideshow was developed to accompany the textbook
• Big Ideas Geometry
• By Larson and Boswell
• 2022 K12 (National Geographic/Cengage)
• Some examples and diagrams are taken from the textbook.
Slides created by
Richard Wright, Andrews Academy
[email protected]
3. 10.1 Lines and Segments that
Intersect Circles
After this lesson…
• I can identify special segments and lines that intersect circles.
• I can use properties of tangents to solve problems.
4. 10.1 Lines and Segments that Intersect Circles
• Circle
• All the points a given distance from a central point in a plane
• Named by the center
• Radius (r) – the distance from the center of the circle to the edge.
• Chord – line segment that connects two points on a circle.
5. 10.1 Lines and Segments that Intersect Circles
• Diameter (d) – chord that goes through the center of the circle (longest chord = 2
radii)
• d = 2r
• What is the radius of a circle if the diameter is 16 feet?
6. 10.1 Lines and Segments that Intersect Circles
• Secant
• Line that intersects a circle twice
• Tangent
• Line that intersects a circle once
7. 10.1 Lines and Segments that Intersect Circles
• What word best describes ?
• What word best describes ?
• Name a tangent and a secant.
• Try #6
8. 10.1 Lines and Segments that Intersect Circles
• Two circles can intersect in 2 points
• 1 point
• No points
9. 10.1 Lines and Segments that Intersect Circles
• Common tangents
• Lines tangent to 2 circles
• How many common tangents do the circles have?
• Try #8
10. 10.1 Lines and Segments that Intersect Circles
Tangent lines are perpendicular to radius.
Tangent segments from the same point are
congruent.
11. 10.1 Lines and Segments that Intersect Circles
• Is tangent to ?
• is a tangent to . Find the value of r.
• Try #16
12. 10.1 Lines and Segments that Intersect Circles
• Find the value of x.
• Try #24
13. 10.2 Finding Arc Measures
After this lesson…
• I can find arc measures.
• I can identify congruent arcs.
14. 10.2 Finding Arc Measures
• How do you cut a pizza into eight equal pieces?
• You cut in half, half, and half
• What measures are the angles in each piece?
• 360 / 8 = 45
15. • There are 360 in a complete circle.
• Central Angle – Angle whose vertex is the center of the circle
• Arcs
• An arc is a portion of a circle (curved line)
• A central angle cuts a circle into two arcs
• Minor arc – smaller of the two arcs – measures of arcs are the measures of
the central angles
• Major arc – bigger of the two arcs
• Named or
• use two endpoints to identify minor arc
• use three letters to identify major arc
10.2 Finding Arc Measures
16. 10.2 Finding Arc Measures
• Name the minor arc and find its measure. Then name the major arc and find its
measure.
• Try #2
17. 10.2 Finding Arc Measures
• Identify as major arc, minor arc, or semicircle. Find the measure.
• Try #8
18. 10.2 Finding Arc Measures
• Semicircle – arc if the central angle is 180
• Similar Circles – all circles are similar
• Congruent circles – same radius
• Congruent arcs – same radius and measure
19. 10.2 Finding Arc Measures
• Tell whether the red arcs are congruent.
• Try #16
20. 10.3 Using Chords
After this lesson…
• I can use chords of circles to find arc measures.
• I can use chords of circles to find lengths.
• I can describe the relationship between a diameter and a chord perpendicular to
a diameter.
21. 10.3 Using Chords
• Chords divide a circle into a major and minor arc.
In the same circle, or circles, two minor arcs are iff
their chords are .
24. 10.3 Using Chords
If one chord is bisector of another chord, then the 1st
chord is diameter.
If a diameter is to a chord, then it bisects the chord
and its arc.
28. 10.4 Inscribed Angles and
Polygons
After this lesson…
• I can find measures of inscribed angles and intercepted arcs.
• I can find angle measures of inscribed polygons.
29. 10.4 Inscribed Angles and Polygons
• What does inscribed mean?
• Writing ON something; engraving ON
• Inscribed angle means the vertex ON the circle.
30. 10.4 Inscribed Angles and Polygons
• Inscribed Angle
• An angle whose vertex is on the edge of a circle and is inside the circle.
• Intercepted Arc
• The arc of the circle that is in the angle.
31. 10.4 Inscribed Angles and Polygons
The measure of an inscribed angle is ½ the measure of the
intercepted arc.
If two inscribed angles of the same or congruent circles
intercept congruent arcs, then the angles are congruent.
32. 10.4 Inscribed Angles and Polygons
If an inscribed angle of a circle intercepts a semicircle, then
the angle is a right angle
½ 180 (semicircle) = 90
If a quadrilateral is inscribed in a circle, then the opposite
angles are supplementary.
33. 10.4 Inscribed Angles and Polygons
• Find the measure of the red arc or angle.
• Try #2
34. 10.4 Inscribed Angles and Polygons
• Find the value of each variable.
• Try #12
• Try #10
35. 10.5 Angle Relationships in
Circles
After this lesson…
• I can identify angles and arcs determined by chords, secants, and tangents.
• I can find angle measures and arc measures involving chords, secants, and
tangents.
• I can use circumscribed angles to solve problems.
36. 10.5 Angle Relationships in Circles
• Find m1
• Try #2
If a secant and a tangent intersect at the point of tangency,
then the measure of each angle formed is one-half the
measure of its intercepted arc.
37. 10.5 Angle Relationships in Circles
• Find the value of x.
• Try #6
If two secants intersect in the interior of a circle, then the
measure of an angle formed is ½ the sum of the measures of
the arcs intercepted by the angle and its vertical angle.
Angles Inside the Circle Theorem
38. 10.5 Angle Relationships in Circles
• What is the value of a?
• Try #10
If two secants, tangents, or one of each intersect in the exterior of a circle,
then the measure of the angle formed is ½ the difference of the measures
of the intercepted arcs.
Angles Outside the Circle Theorem
39. 10.5 Angle Relationships in Circles
• What is the value of x?
The measure of a circumscribed angle is equal to 180°
minus the measure of the central angle that intercepts
the same arc.
Circumscribed Angle Theorem
40. 10.6 Segment Relationships in
Circles
After this lesson…
• I can find lengths of segments of chords.
• I can identify segments of secants and tangents.
• I can find lengths of segments of secants and tangents.
41. 10.6 Segment Relationships in Circles
• A person is stuck in a water pipe with unknown radius. He estimates that surface of
the water makes a 4 ft chord near the top of the pipe and that the water is 6 ft deep.
How much room is available for his head?
4
6
42. 10.6 Segment Relationships in Circles
• Take the example we started above.
• The segments of the horizontal chords are 2 and 2; the segments
of the vertical chords are 6 and x
4
6
If two chords intersect in a circle, then the products of the
measures of the segments of the chords are equal.
Segments of Chords Theorem
43. 10.6 Segment Relationships in Circles
• Find x in the diagram.
If two secants are drawn to a circle from an exterior point, then the
product of the measures of one secant segment and its external secant
segment is equal to the product of the measures of the other secant
segment and its external secant segment.
Segments of Secants Theorem
44. 10.6 Segment Relationships in Circles
• Find x in the diagram
If a tangent segment and a secant segment are drawn to a circle from an
exterior point, then the square of the measure of the tangent segment is
equal to the product of the measures of the secant segment and its
external secant segment.
Segments of Secants and Tangents Theorem
45. 10.7 Circles in the Coordinate
Plane
After this lesson…
• I can write equations of circles.
• I can find the center and radius of a circle.
• I can graph equations of circles.
• I can write coordinate proofs involving circles.
46. 10.7 Circles in the Coordinate Plane
• Equation of a Circle
• Where (h, k) is the center and r is the radius
• Write the equation of the circle in the graph.
• Try #2
47. 10.7 Circles in the Coordinate Plane
• Write the standard equation of the circle.
• Try #8
48. 10.7 Circles in the Coordinate Plane
• Graph a circle by
• Plot the center
• Move every direction the distance r
from the center
• Draw a circle
• Graph
49. 10.7 Circles in the Coordinate Plane
• The point (1, 4) is on a circle centered
at the origin. Prove or disprove that
the point is on the circle.
• Try #19
#41:The chord can be subdivided into 2 ft and 2ft since the vertical line is a diameter. To answer the question we need to know the theorems in the section.
