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Construction of a regular pentagon Dr Andrew French Const. Pent. p1 of 10. A.French 2012

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x y R O ‘Construction’ means one can only draw lines using the following equipment A straight edge A compass Step 1 Draw a circle and divide it vertically and horizontally to form the y and x axes. Use the line bisection method to find the x axis, given the y axis. Const. Pent. p2 of 10. A.French 2012

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O R x y Step 2 Construct a circle with half the radius of the larger circle. Find the centre of this circle by bisecting the OX line as indicated. X Const. Pent. p3 of 10. A.French 2012

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O R x y Step 3. Draw a line from the base of the larger circle and through the centre of the smaller circle. Find where this crosses the upper edge of the smaller circle. Const. Pent. p4 of 10. A.French 2012

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O R x y Step 4. Draw an arc of a large circle with radius from the base of the first circle to the small circle crossing point found in step 3. Find position K where this crosses the original circle. Const. Pent. p5 of 10. A.French 2012 K

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O x y Step 5 Draw a straight line from the intersection of the y axis and the original circle and the crossing point K found in step 4. This is an edge of a regular pentagon! Use a compass to step round the original circle to find the other three points. To minimize the effects of drawing errors, work out the lower vertices from the left and right large circular arc intersections rather than step round using a compass from one side only. Const. Pent. p6 of 10. A.French 2012 K

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O R x y is the –ve solution Const. Pent. p7 of 10. A.French 2012

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O x y Circle theorem Now let’s consider the regular pentagon separately. To demonstrate why the construction works we need an expression for the triangle side k, derived only from properties of the regular pentagon. i.e. independent of the construction. If everything agrees then the construction works! Const. Pent. p8 of 10. A.French 2012

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O x y is the GOLDEN RATIO From the construction By properties of a regular pentagon By equating k we find which indeed is the case! Note Const. Pent. p9 of 10. A.French 2012

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Typically take a > b so GOLDEN RATIO Define ‘Golden rectangle’ by ratio of sides a/b The Golden Ratio Const. Pent. p10 of 10. A.French 2012

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The construction of the regular pentagon therefore gives us another exact right angled triangle relationship From Pythagoras’ theorem

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