The ancient Greeks thought that the most pleasing shape for a rectangle was one for which the ratio of the length to the width was 8 to 5, the golden ratio. If the length of a rectangle painting is 2 ft longer then the width, then for what dimentsions would the length and width have the golden ratio? ( show work please ) :shock:
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The Golden Ratio
- Thread starter sarbie83
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sarbie83 said:The ancient Greeks thought that the most pleasing shape for a rectangle was one for which the ratio of the length to the width was 8 to 5, the golden ratio. If the length of a rectangle painting is 2 ft longer then the width, then for what dimentsions would the length and width have the golden ratio? ( show work please ) :shock:
Please show us your work, indicating excatly where you are stuck - so that we know where to begin to help you.
The ancient Greeks thought that the most pleasing shape for a rectangle was one for which the ratio of the length to the width was 8 to 5, the golden ratio. If the length of a rectangle painting is 2 ft longer then the width, then for what dimentsions would the length and width have the golden ratio? ( show work please )
The golden ratio is not defined by the ratio of 8 to 5. Granted, it is close, but not exactly the true Golden Ratio.
The Golden Ratio is defined by dividing a line segment into two distinct parts such that the ratio of the larger segment to the shorter segment is the same as the ratio of the whole line to the larger segment. What is meant here is to divide a line segment of unit length into two parts. Call the longer segment X and the shorter segment 1 - X. The segments are said to be in the divine proportion if they satisfy the following relationship: 1/X = X/(1 - X). This expression can be restated as X^2 = 1 - X or X^2 + X - 1 = 0. From this X = (-1 + sqrt5)/2 or (-1 - sqrt5)/2. As we cannot recognize a negative length our golden ratio becomes X = (-1 + sqrt5)/2 which computes out to be .618034 and the value of the golden ratio is its reciprocal, 1.610834. (Note that the reciprocal of .618034.... equals 1.618034....!)
Now that we have divided a line segment into the golden ratio, it is possible to construct a golden rectangle. Lets take the line AC with point B located at the golden ratio point such that AC/AB = AB/BC. Construct a square ABED with AB = BE = ED = DA. Construct line CF perpendicular to AC at C and equal to BE. Extend line DE out to point F. Rectangle ACFD is a golden rectangle.
You can also create a golden rectangle without having the golden ratio. Draw any square ABCD, AB on the top and CD on the bottom. Bisect the square vertically with line EF.such that the upper line is now AEB and the lower CFD. With FB as a radius, and F as a center, draw arc BG with G being the point of intersection of the arc on the extension of line CD. Construct a line perpendicular to CG at G to the intersection of the extended line AB at H. Rectangle AHGC is a golden rectangle. If you work your way through the construction algebraically, starting with a unit square, you will end up with the golden ratio of 1.618034.
If you are really interested in the golden ratio, golden rectangle, golden triangle, their application to the world around you, and where you can see examples of their use, you will be able to find much more information in the following references. There are many other mathematics books that offer brief discussions of the Golden Ratio.
Reading References:
1--Mathematics Appreciation by Theoni Pappas, Wide World Publishing/Tetra, San Carlos, CA, 1986.
2--Teaching with Student Math Notes-Volume 2 by Evan M. Maletsky, National Council of Teachers of Mathematics, 1993.
3--The Joy of Mathematics by Theoni Pappas, Wide World Publishing/Tetra, 1989
4--More Joy of Mathematics by Theoni Pappas, Wide World Publishing/Tetra, 1991
5--The Divine Proportion by H.E. Huntley, Dover Publications, Inc.
The golden ratio is not defined by the ratio of 8 to 5. Granted, it is close, but not exactly the true Golden Ratio.
The Golden Ratio is defined by dividing a line segment into two distinct parts such that the ratio of the larger segment to the shorter segment is the same as the ratio of the whole line to the larger segment. What is meant here is to divide a line segment of unit length into two parts. Call the longer segment X and the shorter segment 1 - X. The segments are said to be in the divine proportion if they satisfy the following relationship: 1/X = X/(1 - X). This expression can be restated as X^2 = 1 - X or X^2 + X - 1 = 0. From this X = (-1 + sqrt5)/2 or (-1 - sqrt5)/2. As we cannot recognize a negative length our golden ratio becomes X = (-1 + sqrt5)/2 which computes out to be .618034 and the value of the golden ratio is its reciprocal, 1.610834. (Note that the reciprocal of .618034.... equals 1.618034....!)
Now that we have divided a line segment into the golden ratio, it is possible to construct a golden rectangle. Lets take the line AC with point B located at the golden ratio point such that AC/AB = AB/BC. Construct a square ABED with AB = BE = ED = DA. Construct line CF perpendicular to AC at C and equal to BE. Extend line DE out to point F. Rectangle ACFD is a golden rectangle.
You can also create a golden rectangle without having the golden ratio. Draw any square ABCD, AB on the top and CD on the bottom. Bisect the square vertically with line EF.such that the upper line is now AEB and the lower CFD. With FB as a radius, and F as a center, draw arc BG with G being the point of intersection of the arc on the extension of line CD. Construct a line perpendicular to CG at G to the intersection of the extended line AB at H. Rectangle AHGC is a golden rectangle. If you work your way through the construction algebraically, starting with a unit square, you will end up with the golden ratio of 1.618034.
If you are really interested in the golden ratio, golden rectangle, golden triangle, their application to the world around you, and where you can see examples of their use, you will be able to find much more information in the following references. There are many other mathematics books that offer brief discussions of the Golden Ratio.
Reading References:
1--Mathematics Appreciation by Theoni Pappas, Wide World Publishing/Tetra, San Carlos, CA, 1986.
2--Teaching with Student Math Notes-Volume 2 by Evan M. Maletsky, National Council of Teachers of Mathematics, 1993.
3--The Joy of Mathematics by Theoni Pappas, Wide World Publishing/Tetra, 1989
4--More Joy of Mathematics by Theoni Pappas, Wide World Publishing/Tetra, 1991
5--The Divine Proportion by H.E. Huntley, Dover Publications, Inc.