How to Determine the Mean Proportion or Square Root Geometrically
Understand the terms., Scale up your numbers., Choose factors a and b of 6096 as a=127, b=48 and center = (127+48)/2= 87.5; that's perfectly acceptable., Describe the semicircle above labeled points a,b and c of line ac containing the segments ab...
Step-by-Step Guide
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Step 1: Understand the terms.
Usually, what is meant by a "mean proportional"
is a line (or number) between two lines in length such that its proportion over the first shorter line is the same proportion the second longer line has to it
-- thus it is the mean, And what is meant by "square root" is that number (or line) which when multiplied times itself, produces a square; however, the root itself may be the product or sum of two or more other numbers.
The method presented here takes advantage of exactly that fact, so if you have a square in mind, please make it the product of a number a and a number b. -
Step 2: Scale up your numbers.
If you are finding the square root of a large number, you will need to scale up your centimeter or millimeter ruler, e.g. if your number = 6085 and you want to find the square root of it, then your 100 divisions of a 10 centimeter ruler instead of meaning 100, would seem to need to mean 10,000, so each division is worth 100 more divisions, since 100*100= 10,000.
You would find the number 6096 just shy of
6.1 centimeters then, right? But what you really want to measure in are the factors of 6096, and those will be less than 100 x 100, so let's say each division on your 100 millimeter ruler is worth 10, which gives us the number of 1,000 as our new yardstick. , Mark point a at the far left, point b 127 units to a's right, and point c at the other endpoint, 48 more units to the right, with vertical hash marks.
Also mark the center of the line with a hash mark at
87.5 units. ,,,, And since in the right angle "ADC"
"DB" (or MP) has been drawn from the right angle perpendicular to the base, therefore "DB" is a mean proportional between the the segments of the base, "AB" and "BC".
If you now measure "DB" (or MP), you should find it equals
78.0769, which is not only the square root of 6096 as you originally had sought, and the root of its its factors 127 and 48, but
78.0769 / 48 =
1.6266 and 127 /
78.0769 =
1.6266 as well.
That is to say, the square root is also the mean proportional!,, For more art charts and graphs, you might also want to click on Category:
Microsoft Excel Imagery, Category:
Mathematics, Category:
Spreadsheets or Category:
Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page. -
Step 3: Choose factors a and b of 6096 as a=127
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Step 4: b=48 and center = (127+48)/2= 87.5; that's perfectly acceptable.
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Step 5: Describe the semicircle above labeled points a
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Step 6: b and c of line ac containing the segments ab and bc equaling 127 and 48 respectively.
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Step 7: Draw line "BD" at right angles to line ac up to intersect the circle at point d. Join lines ad and dc.
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Step 8: At point d
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Step 9: write "d/M" and at point b. change it to read "b/P" where "MP" means "Mean Proportional".
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Step 10: Now since angle "ADC" is an angle in a semicircle
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Step 11: it is right.
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Step 12: Final Image
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Step 13: Make use of helper articles when proceeding through this tutorial: See the article How to Determine the Geometric Version of the Golden Mean (Ratio or Proportion) for a list of articles related to Excel
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Step 14: Geometric and/or Trigonometric Art
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Step 15: Charting/Diagramming and Algebraic Formulation.
Detailed Guide
Usually, what is meant by a "mean proportional"
is a line (or number) between two lines in length such that its proportion over the first shorter line is the same proportion the second longer line has to it
-- thus it is the mean, And what is meant by "square root" is that number (or line) which when multiplied times itself, produces a square; however, the root itself may be the product or sum of two or more other numbers.
The method presented here takes advantage of exactly that fact, so if you have a square in mind, please make it the product of a number a and a number b.
If you are finding the square root of a large number, you will need to scale up your centimeter or millimeter ruler, e.g. if your number = 6085 and you want to find the square root of it, then your 100 divisions of a 10 centimeter ruler instead of meaning 100, would seem to need to mean 10,000, so each division is worth 100 more divisions, since 100*100= 10,000.
You would find the number 6096 just shy of
6.1 centimeters then, right? But what you really want to measure in are the factors of 6096, and those will be less than 100 x 100, so let's say each division on your 100 millimeter ruler is worth 10, which gives us the number of 1,000 as our new yardstick. , Mark point a at the far left, point b 127 units to a's right, and point c at the other endpoint, 48 more units to the right, with vertical hash marks.
Also mark the center of the line with a hash mark at
87.5 units. ,,,, And since in the right angle "ADC"
"DB" (or MP) has been drawn from the right angle perpendicular to the base, therefore "DB" is a mean proportional between the the segments of the base, "AB" and "BC".
If you now measure "DB" (or MP), you should find it equals
78.0769, which is not only the square root of 6096 as you originally had sought, and the root of its its factors 127 and 48, but
78.0769 / 48 =
1.6266 and 127 /
78.0769 =
1.6266 as well.
That is to say, the square root is also the mean proportional!,, For more art charts and graphs, you might also want to click on Category:
Microsoft Excel Imagery, Category:
Mathematics, Category:
Spreadsheets or Category:
Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.
About the Author
Joyce Adams
Brings years of experience writing about hobbies and related subjects.
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