Download 14.5 Segment Measures

Geometry – Chapter 14 Lesson Plans
Section 14.5 – Segment Measures
Enduring Understandings: The student shall be able to:
1. Find measures of chords, secants, and tangents.
Standards:
31. Circles
Apples geometric relationships to solving problems, such as relationships between
lines and segments associated with circles, the angles they form, and the arcs they
subtend; and the measures of these arcs, angles, and segments.
Essential Questions: How can we calculate the lengths of the parts of intersecting
secants?
Warm up/Opener:
Activities:
Thm 14-13: If two chords of a circle intersect, then the product of the measures of the
segments of one chord equals the product of the measures of the segments of the other
chord.
B
A
D
E
C
Proof of Theorem 14-13:
Statements
Draw AD and BC forming ABE and
DCE
A  D, B  C
Reasons
Construction
14-2: If inscribed angles intercept the same
arc, then the angles are congruent
AA
CPCTC
Multiplication by common denominator
ABE ~ DCE
AE/DE = EB/EC
AE * EC = ED * EB
Defn: A segment is an External Secant Segment iff it is the part of a secant segment that
is outside a circle.
Thm 14-14: If two secant segments are drawn to a circle from an exterior point, then the
product of the measures of one secant segment and its external secant segment equals the
product of the measures of the other secant segment and its external secant segment.
C
K
L
J
D
Proof of Theorem 14-14:
Statement
Reason
Draw KL and CD forming JCD and JLK Construction
14-2: If inscribed angles intercept the same
C  L
arc, then the angles are congruent
Reflexive property
J  J
AA
JCD ~ JLK
JC/JL = JD/JK
CPCTC
JC * JK = JL * JD
Multiplication by common denominator
A special case of theorem 14-14 is when one segment is a tangent segment. This is
covered in Theorem 14-15: 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
equals the product of the measures of the secant segment and its external secant segment.
E
F
G
Proof of Theorem 14-15:
Statement
Draw EG and EH forming FEG and
FHE
FEG  EHG
F  F
FEG ~ FHE
FE/FH = FG/FE
FE2 = FH * FG
H
Reason
Construction
They intersect the same arc, and by 14-11
if a secant-tangent angle has its vertex on
the circle, then its degree measure equals
one-half the intercepted arc, and by 14-1
the degree measure of an intercepted arc is
one-half the intercepted arc.
Reflexive Property
AA
CPCTC
Multiplication by common denominator
Assessments:
Do the “Check for Understanding”
CW WS 14.5
HW pg 616 - 617, # 8 - 26 all (19) if I only cover this section
HW pg 616 – 617, # 9 – 25 odd (9) if I cover this section with another section