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A Quadratic Linked to Real Contexts

Investigate the Golden Ratio through quadratic equations and its application in real life contexts. Use photo measurements to analyze ratios and find the exact value of the Golden Ratio.

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A Quadratic Linked to Real Contexts

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  1. A Quadratic Linked to Real Contexts

  2. Do these Eyes Have It? What is It?

  3. Photo Measurements • Find the ratio of the distance between the two nearest corners of the eyes to the distance between the corners of the left eye. (AB:BC) • 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. (AC:AB) • What is true about the two ratios?

  4. Too far apart Too far apart Has IT! About right Too close together

  5. What is the value of the ratios when they are equal? Geometer’s 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 ax2+bx+c=0

  6. It is the Golden Ratio Euclid

  7. A B C Golden ratio’s 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 AC:AB = AB:BC “. Activity

  8. A B C Finding the value of the golden ratio 1. Suppose BC=1 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 valueof the golden ratio. 4. Use algebra to find the exact valueof 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? A line segment AC is said to be divided in the golden ratio by B if AC :AB = AB :BC

  9. Real Life Examples of the Golden Ratio

  10. 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. • 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. • What is true about the two ratios? Photo Measurements

  11. 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. • 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. • What is true about the two ratios? Photo Measurements

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