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To divide a given line segment into extreme and mean ratio. |
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To divide a given line segment into extreme and mean ratio. |
| Given: Line segment AB |
| Required: To divide AB into extreme and mean ratio |
| Procedure: |
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 Bisect
AB. (help) or use the midpoint command under the construct menu and find
the midpoint of AB, label it M.. |
Erect a 
at B. (help) or use the
line command under the construct menu and construct a
at B. |
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Use the segment command under the construct menu
and construct segment MB. |
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Use the circle by center + radius command under the construct menu and
construct a circle with center at B and radius congruent to segment MB. |
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Label the intersection of this circle with the perpendicular line point
O. |
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Use the circle by center + radius command under the
construct menu and construct a circle with center O and radius congruent
to segment MB. |
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Use the ray command under the construct menu and
construct ray AO. Label the intersections of ray AO with circle O,
D and E respectively. (Note: ADOE is a secant of circle O) |
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Use the segment command under the construct menu and construct segment
AD. |
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Use the circle by center + radius command under the construct menu and
construct a circle with center at A and radius congruent to segment AD. |
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Label the intersection of this circle with segment AB point C. |
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AB is divided internally at C into extreme and mean ratio. |
| Proof: |
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AE/AB = AB/AD (If from a point outside a circle, a tangent
and a secant are drawn, the tangent is the mean proportional between the
secant and its external segment.) |
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(AE - AB)/AB = (AB - AD)/AD (If 4
quantities are in proportion, they are in proportion by subtraction.) |
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AE - AB = AE - DE = AD =
AC (substitution) |
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AB - AD = AB - AC = BC (substitution) |
 AC/AB
= BC/AC (substitution from above) |
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AB/AC
= AC/BC (If 4 quantities are in proportion, they
are in proportion by inversion) |
| Sketch: |
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This page uses JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright © 1990-2001 by KCP Technologies, Inc. Licensed only for non-commercial use. divide line into mean and extreme ratios |
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