Telescoping triangles
description
Transcript of Telescoping triangles
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TELE
SCOPING
TRIANGLE
S
C O N N I E S
T O N E R & M
Y R A TU R N E R
Illinois
State
Standard
s
7.7.01
Selec
t and
use ap
propri
ate
standa
rd uni
ts and
tools t
o meas
ure
length
, mass
/weight,
capac
ity, an
d
angles
. Sket
ch, with
given
specifi
cation
s, line
segm
ents, a
ngles,
triangl
es, an
d quad
rilater
als.
7.7.03
Compar
e and
estimate
length
(inclu
ding p
erimete
r), are
a, volu
me,
weight/
mass, an
d angl
es (0°
to 18
0°)
using re
ferent
s.
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GIVEN TRIANGLE ∆ABC
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CONSTRUCT MIDPOINTSConstruct a midpoint from each side.
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CONSTRUCT MIDPOINT ∆DEF
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FIND MIDPOINTS OF ∆DEF
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CONNECT MIDPOINTS FORMING ∆GHI
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FIND MIDPOINTS OF ∆GHI
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CONSTRUCT ∆JKL USING MIDPOINTS
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∆ABCSEGMENT LENGTH
∆DEFSEGMENT LENGTH
RATIO ∆ABC/∆DEF
= 11.76 =5.88 2:1 = 20 = 10 2:1 = 16 = 8 2:1∆DEF
SEGMENT LENGTH
∆GHISEGMENT LENGTH
RATIO ∆DEF/∆GHI
= 8 = 4 2:1 = 5.88 = 2.94 2:1
=10 = 5 2:1∆GHI
SEGMENT LENGTH
∆JKLSEGMENT LENGTH
RATIO∆GHI/∆JKL
= 2.94 = 1.47 2:1 = 5 = 2.5 2:1 = 4 = 2 2:1
SEGMENT RATIOS
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Triangle
Area
Triangle
Area
Area Ratios
∆ABC = 94.13
∆DEF = 23.53
∆ABC/∆DEF
4:1
∆DEF =23.53
∆GHI = 5.88
∆DEF/∆GHI
4:1
∆GHI = 5.88
∆JKL =1.47
∆GHI/∆JKL
4:1
TRIANGLE AREA RATIOS
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OBSERVATIO
NS
Midsegment ratios were all 2:1
Area ratios were all 4:1
All triangles similar
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REASONING
Midpoint of each segment ÷ 2, henceArea of ∆ABC = Area of ∆DEF = =
Proving the 4:1 ratio of areas