I am stuck on a question from my history of math textbook. Here is the question: The division of a line segment into two unequal partsso that the whole segment will have the same ratio toits larger part that its larger part has to its smallerpart is called the golden section. A classicalruler-and-compass construction for the golden sectionof a segment AB is as follows. At B erect BC equaland perpendicular to AB. Let M be the midpoint ofAB, and with MC as a radius, draw a semicirclecutting AB extended in D and E. Then the segmentB E laid off on AB gives P, the golden section. (a) Show that 4DBC is similar to 4CBE, whenceDB=BC D BC=B E.(b) Subtract 1 from both sides of the equality inpart (a) and substitute equals to conclude thatAB=AP D AP=P B.(c) Prove that the value of the common ratio in part(b) is (p5 C 1)=2, which is the “golden ratio.”[Hint: Replace P B by AB AP to see thatAB2 AB Ð AP AP2 D 0. Divide thisequation by AP2 to get a quadratic equation inthe ratio AB=AP.](d) A golden rectangle is a rectangle whose sidesare in the ratio (p5 C 1)=2. (The goldenrectangle has dimensions pleasing to the eyeand was used for the measurements of thefacade of the Parthenon and other Greektemples.) Verify that both the rectangles AEFG and BEFC are golden rectangles
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A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
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In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
I am stuck on a question from my history of math textbook. Here is the question:
The division of a line segment into two unequal parts
so that the whole segment will have the same ratio to
its larger part that its larger part has to its smaller
part is called the golden section. A classical
ruler-and-compass construction for the golden section
of a segment AB is as follows. At B erect BC equal
and perpendicular to AB. Let M be the midpoint of
AB, and with MC as a radius, draw a semicircle
cutting AB extended in D and E. Then the segment
B E laid off on AB gives P, the golden section.
(a) Show that 4DBC is similar to 4CBE, whence
DB=BC D BC=B E.
(b) Subtract 1 from both sides of the equality in
part (a) and substitute equals to conclude that
AB=AP D AP=P B.
(c) Prove that the value of the common ratio in part
(b) is (p5 C 1)=2, which is the “golden ratio.”
[Hint: Replace P B by AB AP to see that
AB2 AB Ð AP AP2 D 0. Divide this
equation by AP2 to get a
the ratio AB=AP.]
(d) A golden rectangle is a rectangle whose sides
are in the ratio (p5 C 1)=2. (The golden
rectangle has dimensions pleasing to the eye
and was used for the measurements of the
facade of the Parthenon and other Greektemples.) Verify that both the rectangles AEFG and BEFC are golden rectangles
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