Download Project Golden Ratio - MACMATH-STHS

Geometry
Project: Golden Ratio
Materials:
Straightedge (Centimeters)
Protractor
Compass
You may have heard of golden rectangles and triangles, whose sides have a special
ratio called the golden ratio. The golden ratio is approximately 1.618 to 1. In this project
you will construct and explore golden rectangles and triangles.
Golden Rectangles
Artists and architects often use the golden rectangle because it is pleasing to the eye.
A look at the Parthenon in Greece, the Taj Mahal in India, or the Lincoln Memorial is
Washington, D.C., will reveal uses of the golden rectangle in architecture.
Investigate:
1. Construct a golden rectangle using Diagram 1.
a. Using a compass and a straightedge bisect AD and label the midpoint X.
(textbook page 39)
b. Extend the line containing AD.
c. Using point X as the center and XC as the length draw the arc from point C to
the extension of AD. Label this point F.
d. Using a compass and a straightedge bisect BC and label the midpoint Y.
(textbook page 39)
e. Extend the line containing BC.
f. Using point Y as the center and YD as the length draw the arc from point D to the
extension of BC. Label this point E.
g. Draw the line connecting point F and point E to complete the golden rectangle
ABEF.
2. Find your golden ratio.
a. Find the measure of AF and AB in centimeters.
b. Find the ratio of AF to AB. Express the ratio as a decimal rounded to the
nearest thousandth.
c. How does this compare to the golden ratio (1.618)?
Golden Triangles
Artists often use golden triangles to draw your eye toward the face of a subject. The
folded arms and head of the Mona Lisa from a triangle.
1. Construct a golden triangle using Diagram 2.
a. Draw AC and AD.
b. Find the measure of AC and AD in centimeters.
c. Use a protractor to find the measures of the angles of ADC.
d. Classify ADC. (Right, Isosceles, Equilateral, Obtuse, Acute, Scalene)
2. Find your golden ratio.
a. Find the measure of DC in centimeters.
b. Find the ratio of AD to DC. Express the ratio as a decimal rounded to the nearest
thousandth.
c. How does this ratio compare to the golden ratio?
3. Use your ADC on Diagram 2 to construct another golden triangle.
a. Using a compass and straightedge, bisect ADC. (textbook page 48)
b. Label the point where the angle bisector intersects AC as point F.
c. Find the measure of DF and DC in centimeters.
d. Use a protractor to find the measures of the angles in DCF.
e. Classify DCF. (Right, Isosceles, Equilateral, Obtuse, Acute, Scalene)
f. Find the measure of FC in centimeters.
g. Find the ratio of DC to FC. Express the ratio as a decimal rounded to the nearest
thousandth.
h. How does this ratio compare to the golden ratio?
4. Repeat this process still using Diagram 2.
a. You now have constructed two golden triangles. Use the same process as in step 3
to construct one more golden triangle CFG inside DCF.
b. Classify CFG. (Right, Isosceles, Equilateral, Obtuse, Acute, Scalene)
b. Label all three triangles. You may want to use different colors to outline the
different triangles.
c. Show the golden ratio in each triangle.
5. What conclusion can you make about the classification of the triangles that are golden
triangles?
6. Presenting Your Project:
a. Make a poster to display your golden rectangle and golden triangles.
b. Include labels and ratios.
c. Include pictures of 2 architectural or artistic examples where the golden ratio
appears.
Evaluation:
____1. Construction of rectangle ABEF. (10 points)
____2. Measure of AF and AB. (3 points)
____3. Ratio of AF to AB. (3points)
____4. Comparison of ratio AF:AB to golden ratio. (3 points)
____5. Construction of ADC. (3 points)
____6. Measure of AC and AD. (3 points)
____7. Measures of the angles of ADC. (3 points)
____8. Classification of ADC. (3 points)
____9. Measure of DC. (3 points)
____10. Ratio of AD to DC. (3 points)
____11. Comparison of ratio AD:DC to golden ratio. (3 points)
____12. Construction of DCF. (4 points)
____13. Measure of DF and DC. (3 points)
____14. Measures of the angles of DCF. (3 points)
____15. Classification of triangle DCF. (3 points)
____16. Measure of FC. (3 points)
____17. Ratio of DC to FC. (3 points)
____18. Comparison of ratio DC:FC to golden ratio. (3 points)
____19. Construction of CFG. (5 points)
____20. Conclusion. (3 points)
____21. Poster (20 points)
____22. 2 pictures of architectural or artistic examples of golden ratio. (10 points)