Tangents and Circles in Geometry
Chapter 12.1
Common Core
– G.C.2 Identify and describe
relationships among inscribed angels, radii, and
chords…the radius of a circle is perpendicular to
the tangent where the radius intersects the
circle.
Objective
– To use properties of a tangent to a
circle.
Chapter 12.1 Notes
Circles
– is the set of all pts in a plane that are
equidistant from a given pt, called the center
Internally and Externally Tangents
The
Circles
are
internally and
externally
tangent.
The
Lines
are
internally and
externally
tangent.
Thm
– If a line is tangent to a circle, it is
⊥ to the
radius drawn to the point of tangency
If
then
Thm
– In a plane, if a line is ⊥ to a radius of a
circle at its endpts. on the circle, then the
line is tangent to the circle.
If
then
Thm
– If 2 segments from the same exterior pt
are tangent to a circle, then they are
≌.
If
then
Chapter 12.2 Notes
Common Core
– G.C.2 Identify and describe
relationships among inscribed angles, radii,
and chords.
Objectives
– To use congruent chords, arcs, and
central angles. To use perpendicular bisectors
to chords.
Chapter 12.2 Notes
Central Angle
– an angle made with the center
of the circle.
Minor Arc
– is a central angle less than 180°
Semicircle
– is a central angle that is exactly 180°
Major Arc
– is a central angle more than 180°
Arc Addition Postulate
m ABC = m AB + m BC
Thm
– AB
≌ BC AB ≌ BC
Thm
– DE ≌ EF, DG ≌ GF
Thm
– JK is a diameter of the circle
Thm
– AB
≌ CD EF ≌ EG
Chapter 12.3 Notes
Common Core
–
G.C.2, G.C.3, G.C.4 Identify and
describe relationships among inscribed angles,
radii, and chords. Prove properties of angles
for a quadrilateral inscribed in a circle.
Objectives
– To find the measure of an inscribed
angle. To find the measure of an angle formed
by a tangent and a chord.
Chapter 12.3 Notes
Inscribed Angles
– is an angle whose vertex is on
a circle and whose sides contain chords of the
circle.
Inscribed angle is half the measure of the
intercepted arc.
Thm
– If 2 inscribed angles of a circle intercept
the same arc, then the angles are congruent.
Thm
– If a rt is inscribed in a circle, then the
hypotenuse is a diameter of the circle and vise
versa.
Thm
– A Quadrilateral can be inscribed in a circle
if and only if its opposite angles are
supplementary
The Quad. is
inscribed
in the circle and the
circle is
circumscribed
about the Quad.
Thm
– If a tangent and an chord intersect at a
pt. on a circle, then the measure of each angle
formed is ½ the measure of its intercepted arc.
Chapter 12.4
Common Core
– G.C.2 Identify and describe
relationships among inscribed angles, radii,
and chords.
Objectives
– To find measures of angles formed
by chords, secants, and tangents. To find the
lengths of segments associated with circles.
Chapter 12.4
Thm
– If 2 chords intersect in the interior of a
circle, then the measure of each angle is ½ the
sum of the measures of the arcs intercepted
by the angle and its vertical angles.
Thm
–
If a tangent and a secant, 2 tangents, or 2 secants
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.
Thm
–
If 2 chords intersect in the interior of a circle, then the product of
the lengths of the segments of one chord is equal to the product of the
lengths of the segments of the other chord.
Thm
–
If 2 secant segments share the same endpt outside a circle, then
the product of the length of one secant segment and the length of its
external segment equals the product of the length of the other secant
segment and the length of its external segment
Thm
–
If a secant segment and a tangent segment share an endpt.
Outside a circle, then the product of the length of the secant segment and
the length of its external segment equals the square of the length of the
tangent segment.
Chapter 12.5 Notes
Common Core
– G.GPE.1 Derive the
equation of a circle given center and
radius using the Pythagorean
Theorem.
Objectives
– To write the equation of a
circle. To find the center and radius
of a circle.
Chapter 12.5 Notes
Standard equation of a circle
(x –
h
)
2
+ (y –
k
)
2
=
r
2
Center (
h
,
k
)
Radius is
r
Chapter 12.6 Notes
Common Core
– G.GMD.4 Identify three-
dimensional objects generated by
rotations of two-dimensional objects.
Objectives
– To draw and describe a locus.
Chapter 12.6 Notes
Locus
– is the set of all points in a plane that satisfy
a given condition or a set of given conditions.
Finding a Locus
1)
Draw any figures that are given in the statement of
the problem.
2)
Locate several pts. that satisfy the given condition
3)
Continue drawing pts. Until you can recognize the
pattern.
4)
Draw the locus and describe it in words.
Explore the relationships among inscribed angles, radii, and chords in circles to understand the properties of tangents and their intersections with circles. Learn about the theorems related to tangent lines, segments, and exterior points, and discover the properties of arcs and central angles within circles.
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Presentation Transcript
Chapter 12.1 Common Core G.C.2 Identify and describe relationships among inscribed angels, radii, and chords the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Objective To use properties of a tangent to a circle.
Chapter 12.1 Notes Circles is the set of all pts in a plane that are equidistant from a given pt, called the center
Internally and Externally Tangents The Circles are internally and externally tangent. The Lines are internally and externally tangent.
Thm If a line is tangent to a circle, it is to the radius drawn to the point of tangency If then Thm In a plane, if a line is to a radius of a circle at its endpts. on the circle, then the line is tangent to the circle. If then
Thm If 2 segments from the same exterior pt are tangent to a circle, then they are . If then
Chapter 12.2 Notes Common Core G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Objectives To use congruent chords, arcs, and central angles. To use perpendicular bisectors to chords.
Chapter 12.2 Notes Central Angle an angle made with the center of the circle. Minor Arc is a central angle less than 180 Semicircle is a central angle that is exactly 180 Major Arc is a central angle more than 180
Arc Addition Postulate m ABC = m AB + m BC Thm AB BC AB BC Thm DE EF, DG GF
Thm JK is a diameter of the circle Thm AB CD EF EG
Chapter 12.3 Notes Common Core G.C.2, G.C.3, G.C.4 Identify and describe relationships among inscribed angles, radii, and chords. Prove properties of angles for a quadrilateral inscribed in a circle. Objectives To find the measure of an inscribed angle. To find the measure of an angle formed by a tangent and a chord.
Chapter 12.3 Notes Inscribed Angles is an angle whose vertex is on a circle and whose sides contain chords of the circle. Inscribed angle is half the measure of the intercepted arc.
Thm If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent. Thm If a rt hypotenuse is a diameter of the circle and vise versa. is inscribed in a circle, then the
Thm A Quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary The Quad. is inscribed in the circle and the circle is circumscribed about the Quad.
Thm If a tangent and an chord intersect at a pt. on a circle, then the measure of each angle formed is the measure of its intercepted arc.
Chapter 12.4 Common Core G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Objectives To find measures of angles formed by chords, secants, and tangents. To find the lengths of segments associated with circles.
Chapter 12.4 Thm If 2 chords intersect in the interior of a circle, then the measure of each angle is the sum of the measures of the arcs intercepted by the angle and its vertical angles.
Thm If a tangent and a secant, 2 tangents, or 2 secants intersect in the exterior of a circle, then the measure of the angle formed is the difference of the measures of the intercepted arcs.
Thm If 2 chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Thm If 2 secant segments share the same endpt outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment
Thm If a secant segment and a tangent segment share an endpt. Outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.
Chapter 12.5 Notes Common Core G.GPE.1 Derive the equation of a circle given center and radius using the Pythagorean Theorem. Objectives To write the equation of a circle. To find the center and radius of a circle.
Chapter 12.5 Notes Standard equation of a circle (x h)2+ (y k)2= r2 Center (h,k) Radius is r
Chapter 12.6 Notes Common Core G.GMD.4 Identify three- dimensional objects generated by rotations of two-dimensional objects. Objectives To draw and describe a locus.
Chapter 12.6 Notes Locus is the set of all points in a plane that satisfy a given condition or a set of given conditions. Finding a Locus 1) Draw any figures that are given in the statement of the problem. 2) Locate several pts. that satisfy the given condition 3) Continue drawing pts. Until you can recognize the pattern. 4) Draw the locus and describe it in words.
