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In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem. It is generally attributed to Thales, who is said to have sacrificed an ox in honor of the discovery, but sometimes it is attributed to Pythagoras.
Catigara is also shown at this location on Martin Waldseemueller’s 1507 world map, which avowedly followed the tradition of Ptolemy.[11] Ptolemy’s information was thereby misinterpreted so that the coast of China, which should have been represented as part of the coast of eastern Asia, was falsely made to represent an eastern shore of the Indian Ocean. As a result, Ptolemy implied more land east of the 180th meridian and an ocean beyond. Marco Polo’s account of his travels in eastern Asia described lands and seaports on an eastern ocean apparently unknown to Ptolemy. Marco Polo’s narrative authorized the extensive additions to the Ptolemaic map shown on the 1492 globe of Martin Behaim. The fact that Ptolemy did not represent an eastern coast of Asia made it admissible for Behaim to extend that continent far to the east. Behaim’s globe placed Marco Polo’s Mangi and Cathay east of Ptolemy’s 180th meridian, and the Great Khan’s capital, Cambaluc (near Beijing), on the 41st parallel of latitude at approximately 233 degrees East. Behaim allowed 60 degrees beyond Ptolemy’s 180 degrees for the mainland of Asia and 30 degrees more to the east coast of Cipangu (Japan). Cipangu and the mainland of Asia were thus placed only 90 and 120 degrees, respectively, west of the Canary Islands.
Christopher Columbus modified this geography further by using 53⅔ Italian nautical miles as the length of a degree instead of the longer degree of Ptolemy, and by adopting Marinus of Tyre’s longitude of 225 degrees for the east coast of the Magnus Sinus. This resulted in a considerable eastward advancement of the longitudes given by Martin Behaim and other contemporaries of Columbus. By some process Columbus reasoned that the longitudes of eastern Asia and Cipangu respectively were about 270 and 300 degrees east, or 90 and 60 degrees west of the Canary Islands. He said that he had sailed 1100 leagues from the Canaries when he found Cuba in 1492. This was approximately where he thought the coast of eastern Asia would be found. On this basis of calculation he identified Hispaniola with Cipangu, which he had expected to find on the outward voyage at a distance of about 700 leagues from the Canaries. His later voyages resulted in further exploration of Cuba and in the discovery of South and Central America. At first South America, the Mundus Novus (New World) was considered to be a great island of continental proportions; but as a result of his fourth voyage, it was apparently considered to be identical with the great Upper India peninsula (India Superior) represented by Behaim—the Cape of Cattigara. This seems to be the best interpretation of the sketch map made by Alessandro Zorzi on the advice of Bartholomew Columbus (Christopher’s brother) around 1506, which bears an inscription saying that according to the ancient geographer Marinus of Tyre and Christopher Columbus the distance from Cape St Vincent on the coast of Portugal to Cattigara on the peninsula of India Superior was 225 degrees, while according to Ptolemy the same distance was 180 degrees.[12]
The median is parallel to the bases and its length is the average of the lengths of the bases.
The area of a trapezoid equals the product of the length of the median and the distance between the two parallel sides (which we call the height).
If the angles of a trapezoid along the same base are equal, then the trapezoid is an isosceles trapezoid. In an isosceles trapezoid, the legs are equal in length, as are the diagonals
two general strategies that are useful in trapezoid problems: (1) use what we know about parallel lines and similarity, and (2) drop altitudes to form right triangles.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel (hence the name).
In a parallelogram, opposite sides are equal in length and opposite angles are equal in measure. Also, the diagonals of a parallelogram bisect each other.
The area of a parallelogram equals the product of a side length and the height from that side to the opposite side.
A rectangle is a quadrilateral with all equal angles. Rectangles are parallelograms, inheriting all of those properties. They additionally have equal diagonals.
A rhombus is a quadrilateral with all equal sides. Rhombi are also parallelograms, and their diagonals are perpendicular.
The area of a rhombus, or in fact any quadrilateral with perpendicular diagonals, is half the product of the diagonals.
Regardless of their relative sizes, the bubbles will meet the common wall at an angle of 120 degrees. This is easy to see in the bubble picture to the right. All three bubbles meet at the center at an angle of 120 degrees. Although the mathematics to prove this are beyond the scope of this article, the 120 degree rule always holds, even with complex bubble collections like a foam.
"...If the crests of two or more waves are in step (yellow and magenta waves top), or almost in step, they can combine into a larger or more intense effect (red wave top.) This is called "constructive interference." If the crest of one wave meets the valley of another (yellow and magenta waves bottom), they cancel each other out (red wave bottom.) When two light waves cancel each other, the result is darkness and this is called "destructive interference."
the shortest distance from a point outside a circle to the circle is along the segment connecting the point to the center of the circle.
the sum of the interior angles of a polygon with n sides is 180(n-2) degrees, and • the sum of the exterior angles of a polygon is 360 degrees. • A regular polygon has all its sides equal and all its angles equal. • the measure of each interior angle of a regular polygon is 180(n-2)/n and the measure of each exterior angle is 360/n. • a polygon with n sides has n(n-3)/2 diagonals. • the area of a regular polygon equals half the product of the perimeter of the polygon and the distance from the center of the polygon to one of the sides of the polygon. (This distance is called the apothem.)
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