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Topic: Triangle inequality

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	PlanetMath: triangle inequality (Site not responding. Last check: 2007-10-08)
		Actually, the triangle inequality is one of the properties that define a metric, so it holds on any metric space.
		In planar geometry, this is expressed by saying the each side of a triangle is greater than the difference of the other two.
		This is version 6 of triangle inequality, born on 2002-02-01, modified 2004-04-28.
	planetmath.org /encyclopedia/TriangleInequality.html (187 words)

	Triangle - Gurupedia
		A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.
		A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
		The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base.
	www.gurupedia.com /t/tr/triangle_(geometry).htm (1671 words)

	Illuminations: Inequalities in Triangles
		For example, ask students why it is impossible to have a triangle where the sum of the measures of the small and medium sides is less than the sum of the measure of the large side.
		To begin the inequalities for sides and angles of a triangle part of the lesson, provide students with the Inequalities for Sides and Angles of a Triangle activity sheet and a protractor.
		The inequality for sides and angles of a triangle states that the longest side of the triangle must always be opposite the greatest angle of the triangle and that the shortest side of the triangle must always be opposite the smallest angle of the triangle.
	illuminations.nctm.org /LessonDetail.aspx?ID=L681 (1473 words)

	SparkNotes: SAT Math Level 1: Triangles
		Another property of triangles is that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
		The third important property of triangles is the triangle inequality rule, which states: the length of a side of a triangle is less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
		A 30-60-90 triangle is a triangle with angles of 30º, 60º, and 90º.
	www.sparknotes.com /testprep/books/sat2/math1c/chapter6section2.rhtml (2550 words)

	Triangle inequality - Wikipedia, the free encyclopedia
		In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides.
		The triangle inequality is a theorem in spaces such as the real numbers, all Euclidean spaces, the L
		The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.
	en.wikipedia.org /wiki/Triangle_inequality (412 words)

	Triangle Inequality
		In a neutral geometry, the longest side of a triangle is opposite the angle of greatest measure.
		Therefore, the longest side of a triangle is opposite the angle of greatest measure.//
		In a neutral geometry, the length of one side of a triangle is strictly less than the sum of the lengths of the other two sides.
	www.mnstate.edu /peil/geometry/C3Transform/2TriangleInequal.htm (172 words)

	Triangle Inequality
		We will discuss two commonly used inequality relationships in a triangle: the Triangle Inequality Theorem and the Angle-Side Relationship.
		The Triangle Inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
		In a triangle, the side opposite the larger angle is the longer side.
	www.onlinemathlearning.com /triangle-inequality.html (207 words)

	SparkNotes: Geometric Theorems: Basic Theorems for Triangles
		It states that the length of a side of a triangle is always less than the sum of the lengths of the other two sides.
		Were one side of a triangle longer than the sum of the lengths of the other two, the triangle could not exist.
		A triangle's exterior angle is just like that of any polygon; it is the angle created when one side of the triangle is extended past a vertex.
	www.sparknotes.com /math/geometry2/theorems/section1.html (506 words)

	Triangle Inequality
		I will prove these for you, assuming a and b are real numbers, starting with the first part, the sum, and then using that part to prove the second part, the difference.
		If you can visualize this, then you're more than half-way there, because you know in your heart that the length of the third side of a triangle can't be more than the sum of the lengths of the first two sides.
		There is a very clever proof of the triangle inequality that depends on some interesting properties of the reflection of a complex number about the x-axis.
	mcraefamily.com /MathHelp/BasicArithmeticTriangleInequality.htm (632 words)

	CSUSB Math 129: Triangles
		Make a list of all triangles (by listing the lengths of their sides) that have integer length sides, for all perimeters from 1 through 12.
		Put each triangle on a separate sheet of paper with label and comments, suitable for exhibition.Each person should construct at least one triangle, and the group should use several different construction methods.
		Classify all the triangles constructed by the class in the previous problem according to congruence and similarity.
	www.math.csusb.edu /courses/m129/triangles.html (1133 words)

	Nick's Mathematical Puzzles: Solution 59
		The left side of the inequality is, in fact, true for all triples (a, b, c) of positive real numbers.
		In order to prove the right inequality, we must use the fact that a, b, c are the sides of a triangle.
		The left inequality is known as Nesbitt's Inequality.
	www.qbyte.org /puzzles/p059s.html (262 words)

	v7n4
		Recent reverses of the generalised triangle inequality in normed linear spaces that complement the classical results of Diaz and Metcalf are surveyed.
		In the present note we establish a new integral inequality similar to the Grüss integral inequality by using a variant of the mean value theorem.
		By the application of this result we have obtained a new inequality for the well known means such as arithmetic, geometric and harmonic.
	rgmia.vu.edu.au /v7n4.html (910 words)

	Measuring Up: Triangle Inequality Property
		Repeat this process using angle vertex points that are approximately 2 inches from the left endpoint and 4 inches from the right endpoint.
		Then explain in your own words what must be true about the lengths of the 2 end segments of the pipe cleaner in order to be sure that they will produce a triangle.
		Suppose someone created a pipe cleaner triangle that looked like the one illustrated below, and that when they measured the lengths of a, b, and c they found that the length of a + the length of b was equal to the length of c.
	www.math.niu.edu /mathed/measup1998/triangle.html (213 words)

	The triangle inequality and geometric transformations
		Often the triangle to which we must apply the triangle inequality does not appear in the diagram for the problem.
		Prove that the perimeter of triangle ABC is not less than twice the distance OC, where O is the vertex of the given right angle.
		Prove that the sum of the distances from point O to the vertices of a given triangle is less than the perimeter, if point O lies inside the triangle.
	www.math.uci.edu /~mathcirc/math194/lectures/ineq/node3.html (352 words)

	Technology Center of DuPage - Triangle Area Calculator (Site not responding. Last check: 2007-10-08)
		The typical formula for calculating the area of a triangle is 1/5(Base)(Height) which many people describe as one-half the base of the triangle* times the height.
		A formula for calcualting the area of a triangle when all sides are known is attirbuted to two famous mathematicians; Heron of Alexandria and Archimedes.
		The triangle inequality theorem basically says that all three sides of a triangle have to meet.
	www.mste.uiuc.edu /dildine/tcd_files/program17.htm (263 words)

	Types of Triangles
		The first relationship involves the lengths of the sides of a triangle.
		The second relationship involves the lengths of the sides of a triangle in relation to the triangle's angles.
		Suppose we want to know which side of this triangle is the longest.
	regentsprep.org /Regents/math/triineq/LTriIneq.htm (263 words)

	Triangle inequality
		Although it is easily motivated, triangle inequality is extremely useful.
		Note that in solving Problem 5, we used a slight generalization of the triangle inequality (quadrilateral inequality).
		Prove that the distance between any two points inside a triangle is not greater than half the perimeter of the triangle.
	www.math.uci.edu /~mathcirc/math194/lectures/ineq/node2.html (512 words)

	Triangle Congruence Theorems
		Remember, though, that experiments with specific triangles are not a substitute for a proof for all triangles.
		The Triangle Inequality and the Side-Side-Side Triangle Congruence Theorem.
		Use a ruler and protractor, and the technique in the SAS diagram to either construct a triangle with the given meausrements.
	www.math.csusb.edu /courses/m129/tri_congr.html (1000 words)

	PlanetMath: ultrametric triangle inequality (Site not responding. Last check: 2007-10-08)
		in the ultrametric triangle inequality gives the (*) as result.
		Cross-references: inequality, limit, root, bounded, unity, multiples, archimedean, isosceles, vertices, triangle, implies, ultrametrics, valuation, metrics, non-archimedean, Krull valuation, postulates, satisfies, function, ordered group equipped with zero, field
		This is version 17 of ultrametric triangle inequality, born on 2004-12-16, modified 2005-06-22.
	planetmath.org /encyclopedia/UltrametricTriangleInequality.html (185 words)

	Triangle inequality (Site not responding. Last check: 2007-10-08)
		In mathematics, the triangle inequality states that the distance from A to B to C is never shorter than going directly from A to C. The Triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, L
		In the usual Minkowski space and in Minkowski space extended to an arbitrary number of spatial dimensions, assuming null or timelike vectors in the same time direction, the Triangle inequality is reversed: : ''x'' + y
		≥ 0 A physical example of this inequality is the twin paradox in special relativity.
	triangle-inequality.iqnaut.net (251 words)

	The Triangle Inequality (Site not responding. Last check: 2007-10-08)
		Formulate a relationship between the longest side and two shorter sides of a triangle that is indicated by the data of your table.
		With the numbers of your partner, for each set of numbers draw a triangle whose sides have lengths equal to the three numbers in the set, or determine if no such triangle can be drawn.
		Describe, given three numbers, how to draw a triangle whose sides have lengths equal to the three numbers.
	www.math.nmsu.edu /~pmorandi/math112f00/TriangleInequality.html (175 words)

	Pedoe's Two Triangle Inequality
		Equality occurs iff the two triangles are similar.
		The area of triangle a', b', c' is
		This is equivalent to the codition that the triangles abc and a'b'c' are similar.
	www.physicsforums.com /showthread.php?t=75914 (639 words)

	Keymath.com
		This sketch shows triangle ABC, the lengths of the sides, AB, AC, and CB, and the sum AC + CB.
		Formulate the Triangle Inequality Conjecture: The sum of the lengths of any two sides of a triangle is _____ the length of the third side.
		It is often said that the shortest distance between two points is along the segment connecting them.
	www.keymath.com /x3323.xml (205 words)

	Sketchpad and the triangle inequality theorem (Site not responding. Last check: 2007-10-08)
		People who are unfamiliar with the Triangle Inequalty Theorem will say either 4 miles or 12 miles and not realize what they are stating.
		The Triangle Inequalty Theorem states that one side of any triangle has to be less than the sum of the other two sides of the triangle.
		This is an excellent assignment for the students to practice their skills for Sketchpad.
	www.msu.edu /~mitch321/triangineq.htm (447 words)

	Scalene Inequality / The Geometry of ROTC (Site not responding. Last check: 2007-10-08)
		In ΔABC (formed by the engineer tape connecting cadets A, B, and C together) it is given that the distance of AC > AB.
		Assemble cadet D in between the engineer tape found between cadets A and C such that AD = AB, and extend engineer tape from cadet B to D. By properties of betweeness, m∠ABC > m∠ABD, and by the Isosceles triangle theorem, m∠ABD = m∠ADB.
		Case 2: The engineer tape opposite the angle with the greater measure is the longer side.
	www.plu.edu /~alarcorn/scalene-inequality/home.html (150 words)

	Generalizations of the Triangle Inequality (Site not responding. Last check: 2007-10-08)
		Since the 8th General Inequalities meeting in Hungary (September 15-21, 2002), the author has been considering an idea that as triangle inequality, the inequality
		Triangle inequality, Hilbert space, Sum of two Hilbert spaces, various operators among Hilbert spaces, Reproducing kernel, Linear mapping, Norm inequality.
		Report of the General Inequalities 8 Conference; September 15-21, 2002, Noszvaj, Hungary
	www.maths.tcd.ie /EMIS/journals/JIPAM/v4n3/140_02.html (235 words)

	v7(E)
		Quadratic Reverses of the Continuous Triangle Inequality for Bochner Integral of Vector-Valued Functions in Hilbert Spaces
		Reverses of the Continuous Triangle Inequality for Bochner Integral of Vector-Valued Functions in Hilbert Spaces
		Reverses of the Continuous Triangle Inequality for Bochner Integral of Vector-Valued Functions in Banach Spaces
	rgmia.vu.edu.au /v7(E).html (341 words)

	Untitled Document (Site not responding. Last check: 2007-10-08)
		-The triangle inequality theorem: the sum of the lengths of any two sides of a triangle is greater than the length of the third.
		The sum of the lengths of ______ _______ sides of a triangle is _________ than the length of the ______side.
		5) Now, highlight a vertex on the triangle and move it around the screen, pay attention to the measurements of the sides while you do this, especially the comparison between side AC and the sum of sides AB and BC.
	www.auburn.edu /~jordarl/act1.htm (570 words)

	Mathematics Geometry Homework Help
		Real Analysis - verify the triangle inequality in the special cases where: 1-a and b have the same sign 2-a>=0, b
		Proof - If O is an interior point of triangle ABC prove that OA+OB+OC a- is greater than 1/2 (perimeter of triangle ABC) b- is less than the perimeter of triangle ABC
		- Let d be a metric in X. Prove that p(x,y)=(d(x,y))/(1+d(x,y)) is also a metric in X. 1 problem - Triangle Inequality Theorem: Write a compound inequality to describe tha range of possible measures for side c in terms of a and b.
	www.brainmass.com /homeworkhelp/math/geometry/9070 (202 words)

	A geometric inequality (Site not responding. Last check: 2007-10-08)
		Let ABC be an equilateral triangle and P a point in the plane.
		Proof: Erect equilateral triangle APD, with the angles
		Also, AB=AC and AP=AD, so by side-angle-side the triangles ABP and ACD are congruent.
	www.math.umbc.edu /~rouben/Geometry/inequality.html (145 words)

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