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.

GIVEN TRIANGLE ∆ABC

CONSTRUCT MIDPOINTSConstruct a midpoint from each side.

CONSTRUCT MIDPOINT ∆DEF

FIND MIDPOINTS OF ∆DEF

CONNECT MIDPOINTS FORMING ∆GHI

FIND MIDPOINTS OF ∆GHI

CONSTRUCT ∆JKL USING MIDPOINTS

∆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

 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

OBSERVATIO

NS

Midsegment ratios were all 2:1

Area ratios were all 4:1

All triangles similar

REASONING

Midpoint of each segment ÷ 2, henceArea of ∆ABC = Area of ∆DEF = =

Proving the 4:1 ratio of areas