CIRCLES: Chapter 11
All sketchpad documents on the shared drive in the Santucci folder for Circles.
Open the “Circle Facts” sketchpad document and click on the “DEFINITIONS” tab.
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DEFINE in your notebook and name an e xample of each
(use correct notation):
1. Circle
2. Radius
B
3. Diameter
4. Chord
5. Tangent line
6. Secant line
C
7. Point of tangency
8. Centra l angle
A
9. Insc ribed angle
D
10. Minor a rc
11. Ma jor arc
12. Semicircle
E
Radius of a circle (p. 386) A radius of a circle is any segment with one endpoint
on the circle and the other endpoint at the center of the circle. Radius can also
mean the length of this segment
Diameter of a circle (p. 386) A diameter of a circle is a segment that contains the
center of the circle and whose endpoints are on the circle. The term diameter can
also mean the length of this segment.
Chord (p. 590) A chord of a circle is a segment whose endpoints are on the
circle. The diameter is the longest Chord in a circle
Secant (p. 607) A secant is a line, ray, or segment that intersects a circle at two
points.
Central angle of a circle (p. 386) A central angle of a circle is an angle whose
vertex is the center of the circle.
Inscribed angle (p. 598) 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.
Minor arc (p. 387) A minor arc is smaller than a semicircle.
Major arc (p. 387) A major arc of a circle is an arc that is larger than a
semicircle.
Semicircle (p. 387) A semicircle is half a circle.
Circle Theorems & Postulates: Summarize all questions from this handout in your notebook.
All theorems and postulates can be found in Chapter 11.
From Circle Facts sketchpad document:
1. Two tangents
2. Two chords
3. Perpendicular bisector of chords
4. Central and inscribed angles
From Other sketchpad documents:
5. Angles in a circle:
 Chords that intersect inside a circle
 Chords that intersect outside a circle
6. Segments in a circle:
 Chords that intersect inside a circle
 Chords that intersect outside a circle
CIRCLE FACTS: TANGENT LINES, CHORDS, AND ARCS
Open “Circle facts” sketchpad document
Part 1: Two tangents
1. Select the “two tangents” tab.
B
2. Select the radius segment and point A (where this radius intersects the circle).
A
3. Construct perpendicular line.
4. Select the other radius segment and point B then construct perpendicular line.
5. You should have constructed two tangent lines. Move points around to make sure that
the lines remain tangent to the circle. What is true about the radius and tangent line at the
point of tangency?
6. Select both lines, construct intersection and label it P.
7. Select P and A then measure distance. Select P ang B then measure distance.
8. What is true about the length of two tangent segments from the same point outside the
circle?
A
Part 2: Two chords within a circle
D
G
C
H
B
1. Select the “two chords” tab
E
2. Move points until EH = AB then measure CH and CG (this is the perpendicular distance
from C to each chord). What is true about congruent chords and their distance from the
center of a circle?
3. Move the chords again but make sure that they are equidistant from C. What is true
about two chords equidistant from the center of a circle?
4. Make sure EH = AB. Select arc ED and measure arc angle. Select arc AB and measure
arc angle. What is true about the arc measures of two  chords?
5. Keep the measure of arc ED = measure of arc AB. Measure <ACB and <DCE. What is
true about the central < measures which cut off  arcs?
A
Part 3 Perpendicular bisector of chords:
1. Construct the perpendicular bisector of chord AB (i.e. construct the midpoint and a
perpendicular line through it).
B
2. Move points around. What do you notice about this perpendicular bisector? What do
we call the chord it creates on the circle?
3. Complete each of the following:
a.) A diameter that bisects the chord is __________________________.
b.) A chord that contains the center of the circle ___________ a chord of the
circle.
c.) A radius that is ______________ to a chord bisects the chord.
D
Part 4 Inscribed angles:
R
C
1. Click on the “inscribed angles” tab. <ADC & <ARC are called __________________
A
angles
B
2. The measure of <ADC & < ARC = _____________ the measure of its intercepted arc.
3. If two <’s intercept the same arc then they are ____________________ .
4. Move points so that arc AC measures 180. An angle inscribed in a semicircle is always
D
a
___________ angle.
C
A
B
5. Measure <FHE and <FGE in the 2nd circle. Move points around and observe. Opposite
angle of
any quadrilateral inscribed in a circle are always _____________________________ .
CIRCLES AND SEGMENT LENGTHS
Theorem: For a given point and circle, the product of the lengths of the two segments
from the point to the circle is constant along any line through the point and circle.
1.
THEOREM says rule is…
Chords that intersect INSIDE a circle.
Measure and calculate a*b and c*d
d
b
a
c
2.
Chords that intersect OUTSIDE a circle.
Measure and calculate
w*(w+x)
y*(y+z )
x
w
z
y
3.
Chords that intersect OUTSIDE a circle.
Measure and calculate t2 and y*z
z
y
P
t
ANGLES FORMED BY TWO LINES INTERSECTING INSIDE AND OUTSIDE OF A CIRCLE
1. Discover the Theorem about angles formed by two lines that intersect inside a
circle
Chords that intersect INSIDE a circle.
Find the average of intercepted arcs DE and AC compare with m<ABC
D
A
m DE on
B
FC = 117.41
m AC on
F
FC = 77.51
E
C
2. Discover the Theorem about angles formed by two lines that intersect outside a
circle
a.)
Angle formed by two lines that intersect OUTSIDE a circle.
Find mJC - mGI and compare with m<GHI
J
G
F
H
I
C
b.)
Angles formed by two chords intersecting OUTSIDE the circle.
Calculate the difference mHJ - mJH . How does it compare to m<JSH
J
I
H
S
c.)
Angles formed by two chords intersecting OUTSIDE the circle.
Calculate the difference mHN - mMH . How does it compare to m<MOH
N
I
M
H
O
3. Read theorem 11-11on p. 607 and the proof on p. 608. Copy it into your
notes.