Transcript Geometry and Scale: Reasoning and Proof at Work

Geometry and Scale:
Reasoning and Proof at Work
Problem to think about
An isosceles triangle of
area 100 in2 is cut by a line
parallel to its base into an
isosceles trapezoid and a
smaller isosceles triangle.
The trapezoid has area 75
in2.
Question
If the altitude of the triangle
is 20 in, what is the length
of the “cut line”?
A
D
?
C
E
B
MathCounts, 2004
Problem to think about
• Possible approaches?
– Compute lengths of all
sides of ABE, use similar
triangles, …
– Use formulas for areas of
triangles and trapezoid
– Use proportions and
scaling formulas for
lengths and areas
• Let’s start with simpler
questions
A
D
?
C
E
B
MathCounts, 2004
Scaling Distances
• Think about maps at
different scales
• At 1 in : 1 mile, a 10
mile straight road
measures 10in.
• How long is that road
on a 1 in : 5 miles
scale map?
• How long is that road
on a 1 in : 2 miles
scale map?
Scaling Distances & Areas
• What would be the linear
scale factor if the area of
the map is to be halved?
– 4 units wide becomes
approx 2.8 units
• By what factor would you
increase the sides of a
square in order to double
its area?
• What happens to the
area if we double the
side lengths?
http://www.mapquest.com/
Scaling Lengths & Areas
• By what factor would you • Let’s make a table
increase the radius of a
• What happens with
circle in order to double
circles, squares?
its area?
• What about doubling the
circumference?
Spreadsheet: Areas.xls
• Rectangles – TV
screen, HDTV?
• Parallelogram?
• Trapezoid?
• Triangle?
Parallelogram
Parallelogram
Parallelogram
Area = Base x Height
What happens if we scale the whole parallelogram
keeping proportions?
Areas scale like
Square
of length scale factor
What does this mean for our
original problem?
Problem to think about
An isosceles triangle of
area 100 in2 is cut by a line
parallel to its base into an
isosceles trapezoid and a
smaller isosceles triangle.
The trapezoid has area 75
in2.
Question
If the altitude of the triangle
is 20 in, what is the length
of the “cut line”?
A
D
?
C
E
B
MathCounts, 2004
Data
Area of triangle
1
  Base  Height 
2
Area ABE  100in
2
A
D
?
C
E
Height  20in
Area ACD  100  75  25in
2B
MathCounts, 2004
A
Area ABE
Area ACD 
4
Length BE
Length CD 
2
?
D
C
E
B
Length BE
 Height  Area ABE
2
2  Area ABE 2 100
Length BE 

 10
Height
20
Length BE 10
Length CD 

5
2
2
Summary
• Review of basic area
formulas
• Effect of scaling on
length and area
• Squaring numbers
• Power of mathematical
reasoning
• A little thought can save
a lot of detailed difficulty
• Note: we are still
“problem-solving”
Content Strands
• Geometry, and
• Algebra
• Number sense &
operations
• Measurement
Process Strands
• Reasoning & proof,
and
• Problem solving
• Connections (maps)
• Others?