A Quadratic Linked to Real Contexts

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A Quadratic Linked to Real Contexts
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Do these Eyes Have It?
What is It?
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Photo Measurements

1. Find the ratio of the distance between the two
   nearest corners of the eyes to the distance
   between the corners of the left eye. (ABBC)
2. Find the ratio of the distance between the corner
   of the right eye and the farthest corner of the
   left eye to the distance between the two nearest
   corners of the eyes. (ACAB)
3. What is true about the two ratios?

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Too far apart
Too far apart
Has IT!
About right
Too close together
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(No Transcript)
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What is the value of the ratios when they are
equal?

• Geometers Sketchpad can help find the answer.
• Change eye distances to make ratios equal
• Use graphs of the ratios
• Use a QUADRATIC EQUATION based on ratios

ax2bxc0
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It

• is the
• Golden Ratio

Euclid
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Golden ratios origins

• Although Euclid (300 BC) did not use this term,
  we can find a geometrical reference to the golden
  ratio in his book The Elements
• A line AC is said to have been divided in the
  golden ratio by B if ACAB ABBC .

Activity
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Finding the value of the golden ratio
A line segment AC is said to be divided in the
golden ratio by B if AC AB AB BC

• 1. Suppose BC1 and AB x. Write the ratio
  equation in terms of x.
• 2. To find the value of the golden ratio using
  algebra, what quadratic equation would you have
  to solve?
• 3. Using your calculator, solve this equation
  graphically to find an approximate value of the
  golden ratio.
• 4. Use algebra to find the exact value of the
  golden ratio by solving the equation in (2).
  Solve the equation in two different ways
• (i) complete the square to help factor the
  quadratic and then use the Null Factor Law
  
• (ii) any other algebraic method
• 5. Do your answers in (3) and (4) give the same
  value for the golden ratio as that found by use
  of GSP?

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Real Life Examples of the Golden Ratio
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1. Find the ratio of the distance from the top of
   head to bottom of jacket to the distance from
   bottom of jacket to bottom of dress.
2. Find the ratio of the distance from top of head
   to bottom of dress to the distance from top of
   head to bottom of jacket.
3. What is true about the two ratios?

Photo Measurements
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1. Find the ratio of the distance from the top of
   head to bottom of red colouring to the distance
   from bottom of red to bottom of blue colouring.
2. Find the ratio of the distance from top of head
   to bottom of blue colouring to the distance from
   bottom of red to bottom of blue colouring.
3. What is true about the two ratios?

Photo Measurements