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= 1.5:1 Dividing each side of the ratio by 2
This means that for every 1 rotation of the larger wheel the smaller wheel rotates
1.5 times. Note the way the ratio is reversed, because the smaller wheel turns faster, so the smaller gear wheel is turning 1.5 times faster than the larger gear wheel.
GEOMETRY THEOREMS 215
If the gear wheels are not connected by a chain, but the teeth mesh together (see figure b), then the ratio of wheel turns is the same as if connected by a chain, but the wheels turn in opposite directions.
References: Difference Tables, Patterns.
Here is a list of all the geometry theorems, with their references, which are listed in this book. To find them search under their title or their references.
♦ Complementary angles add up to 90°
♦ Supplementary angles add up to 180°
♦ Conjugate angles add up to 360°
♦ Sum of adjacent angles = 180° (see Adjacent Angles)
♦ Alternate angles are equal.
♦ Sum of angles at a point = 360° (see Angles at a Point)
♦ Sum of angles of a triangle = 180° (see Angle Sum of a Triangle)
♦ Base angles of an isosceles triangle are equal (see Isosceles Triangle)
♦ Sum of cointerior angles = 180° (see Alternate Angles)
216 GOLDEN RATIO
♦ Corresponding angles are equal (see Alternate Angles)
♦ Exterior angle of a triangle = sum of interior opposite angles (see Exterior Angle of a Triangle)
♦ Sum of exterior angles of a polygon is 360° (see Exterior Angle of a Polygon)
♦ Vertically opposite angles are equal.
♦ Angle at the center is twice the angle at the circumference
♦ Angles on the same arc are equal
♦ Angle in the alternate segment
♦ Sum of opposite angles of a cyclic quadrilateral = 180° (see Cyclic Quadrilateral)
♦ Angle in a semicircle is a right angle
♦ Exterior angle of a cyclic quadrilateral = interior opposite angle (see Cyclic Quadrilateral)
♦ Radius is perpendicular to tangent (see Tangent and Radius Theorem)
♦ Tangents from a common point are equal
Tests for Coneyclic Points
♦ Converse of the angle in a semicircle is a right angle (see Angle in a Semicircle)
♦ Converse of sum of opposite angles of cyclic quadrilateral = 180° (see Cyclic
♦ Converse of angles on the same arc are equal (see Cyclic Quadrilateral) Chords and Tangents
♦ Intersecting chords inside a circle
♦ Intersecting chords outside a circle
♦ Tangent secant theorem (see Intersecting Chords)
This is also known as the golden rectangle. Suppose a line segment AB is divided up into two unequal parts by the point C (see figure a). The point C divides the line segment in the golden ratio if the ratio of the complete segment AB to the larger segment AC is equal to the ratio of the larger segment AC to the smaller segment CB. This statement of the golden ratio can be expressed as
AB AC AC ~ CB
GOLDEN RATIO 217
A 1 C X B
Suppose the longer part AC is of length 1 unit, and the shorter part CD is of length x units. The golden ratio can now be expressed as follows:
1 + x 1
AB = 1 + x, AC= 1, CB = x
x(l + x) = 1 Cross multiplying
x + x2 = 1 Expanding the brackets
x2 + x — 1 =0 Rearranging into quadratic form
—b ± \/b2 — 4ac
1 ± t/12 —4(1)(—1) 2 x 1
Formula for solving a quadratic equation Substituting a = 1, b = l,c = — 1
x = 0.618 (to 3 dp) Discarding the negative solution of the
What this value for x means is that if a rectangle has a length of 1 unit and a width of 0.618 unit, then it is a golden rectangle and its sides are in the golden ratio (see figure b). The value x = 0.618 is approximately equal to the fraction 21/34.
An enlargement of a golden rectangle is also a golden rectangle. For example if the rectangle of length 1 and width 0.618 is enlarged three times to obtain a rectangle of length 3 units and width 1.854 units, then this rectangle is also a golden rectangle.
218 GOLDEN RATIO
It is also true that a golden rectangle can have a length of 1.618 units and a width of 1 unit (see figure c), because when the width 1 is divided by 1.618 the result is
Example. Test the dimensions of an ordinary playing card to see if it is a golden rectangle (see figure d).
Solution. The length of the playing card is 8.85 cm and the width is 5.75 cm. Write
Width 5.75 ^ ,
= —- Finding the ratio width: length
= 0.649 (to 3 dp)
The playing card is not quite a golden rectangle, because the golden ratio of width: length should be 0.618, to 3 dp.
Another property of the golden rectangle is that if the rectangle has a square removed with dimensions equal to the shorter side of the rectangle, then the rectangle that remains is itself a golden rectangle. The rectangle that remains will of course not have lengths of 1 unit and 0.618 unit, but the ratio of length:width will be equal to the ratio 1:0.618. This process can be repeated over and again, as demonstrated in figure e.