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Topic: Perpendicular bisector

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	Bisection - Wikipedia, the free encyclopedia
		Particularly important is the perpendicular bisector of a segment, which, according to its name, meets the segment at right angles.
		The perpendicular bisector of a segment also has the property that each of its points is equidistant from the segment's endpoints.
		The exterior bisector of an angle is the line or line segment that divides it into two equal angles on the opposite side as the angle.
	en.wikipedia.org /wiki/Bisection (487 words)

	Perpendicular - Wikipedia, the free encyclopedia
		Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B. If a line is perpendicular to another as in Figure 1, the two angles created are called right angles, or angles measuring 90°.
		Step 1 (red): construct a circle with center at P to create points A' and B' on the line AB, which are equidistant from P. Step 2 (green): construct circles centered at A' and B', both passing through P. Let Q be the other point of intersection of these two circles.
		To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal.
	en.wikipedia.org /wiki/Perpendicular (360 words)

	Glencoe Mathematics - Online Study Tools
		The segment that bisects an angle of the triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle is called the _________.
		A _________ is a line or segment that passes through the midpoint of a side of a triangle and is perpendicular to that side.
		The __________ is the point of concurrency of the angle bisectors of a triangle.
	www.glencoe.com /sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-829637-4&chapter=5&lesson=1&quizType=1&headerFile=4&state= (104 words)

	Math Forum - Geometry Problem of the Week (Site not responding. Last check: 2007-10-24)
		Because the center of the circle is equidistant from the two endpoints of the chord, it is on the perpendicular bisector of the chord.
		The perpendicular bisector of AB consists of all points that are equidistant from A and B and includes P. (This derives from the standard method of constructing a perpendicular bisector or we can prove it by showing that the two right triangles APQ and BPQ are congruent because of SSS).
		F AFBD is a circle AB is a chord with endpoints A and B. FD is the perpendicular bisector of AB.
	mathforum.org /geopow/fullsolutions/053097.fullsolution.html (3061 words)

	Geometry Session 1: Solutions
		This new line segment is perpendicular to the original one and bisects it, because we used the same process that we used to construct the midpoint in the sample construction.
		Construct a perpendicular bisector of the chosen side using the construction from Problem C1(c).
		Another is to draw the perpendicular bisectors of two consecutive sides of the square [the same construction as Problem C6(a)].
	www.learner.org /channel/courses/learningmath/geometry/session1/solutions_c.html (1135 words)

	Proof of the three perpendicular bisectors of the sides of a triangle are concurrent.
		Proof of the three perpendicular bisectors of the sides of a triangle are concurrent.
		Furthermore, K is taken to be the intersection of the perpendicular bisectors p and q.
		Hence, we have proved that the three perpendicular bisectors of the sides of a triangle are concurrent.
	jwilson.coe.uga.edu /EMT668/EMT668.Folders.F97/Anderson/finalprojectpart2/finalprojectpart2proof.html (406 words)

	Math Forum - Geometry Problem of the Week
		Since the perpendicular bisector of any chord goes through the center of the circle, a polygon can be circumscribed by a circle only if the perpendicular bisectors of each side of the polygon converge at a single point, or the midpoint of the circle (see figure X).
		I tried drawing perpendicular bisectors to each side of a polygon and found that if they all meet at one point, then a single circle can be drawn with its centre at that point, which will circumscribe the whole polygon, i.e.
		Hence, using the previous part of the problem, the perpendicular bisectors of the sides of the polygon would all pass through the point O. So, one way to test would be to construct all the perpendicular bisectors of the sides and see if they all meet at one point.
	mathforum.org /pow/solutio59.html (1288 words)

	Construction of the Perpendicular Bisector (Site not responding. Last check: 2007-10-24)
		The perpendicular bisector of a line segment AB is one of the most common and elementary point loci in geometry.
		The perpendicular bisector of a line segment AB is the line through the midpoint (E in the applet below) perpendicular to AB.
		Three perpendicular bisectors of the sides of a triangle intersect in the triangle's circumcenter.
	www.cut-the-knot.org /Outline/Geometry/PerpBisector.shtml (473 words)

	Circles Theorem 6 (Site not responding. Last check: 2007-10-24)
		The perpendicular bisectors of the three sides of a triangle all meet at the center of the circumscribed circle.
		Let O be the point where the perpendicular bisector of AB meets the perpendicular bisector of AC.
		Thus O is on the perpendicular bisectors of all three sides, and since it is equidistant from all three points of the triangle, all three points of the triangle will lie on the same circle centered about O.
	www.sonoma.edu /users/w/wilsonst/courses/math_150/Theorems/Circles/C6.html (74 words)

	Conjcinq
		C-37 In an isosceles triangle, the bisector of the vertex angle is also the attitude to the base and the median to the base (Vertex Angle Bisector Conjecture).
		C-42 The diagonals of a kite are perpendicular (Kite Diagonals Conjecture).
		C-45 The vertex angles of a kite are bisected by a diagonal (Kite Angle Bisector Conjecture).
	members.fortunecity.com /mathman55/hw/conjcinq.htm (1234 words)

	MATHEMATICS DICTIONARY
		In either case, the perpendicular bisector is the set of all points equidistant from the end points of the segment.
		The condition (in analytic geometry) that a line be perpendicular to a plane is that its direction numbers be proportional to those of the normal to the plane; or, what amounts to the same thing, that its direction numbers be proportional to the coefficient of the corresponding variables in the equation of the plane.
		Two perpendicular planes are two planes such that a line in one, which is perpendicular to their line of intersection, is perpendicular to the other; i.e., planes forming a right dihedral angle.
	lines-and-dots.org /perpe.html (417 words)

	Perpendicular Bisectors (Site not responding. Last check: 2007-10-24)
		The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
		In this illustration, segment AB is bisected by line CD intersecting it at its midpoint E. According to the theorem, point F, which is on the perpendicular bisector, should be equidistant to points A and B at all times.
		The perpendicular bisector converse theorem, on the other hand, states that if a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.
	marian.creighton.edu /~dkath/PerpendicularBisector.htm (193 words)

	Transitivity in Action (Site not responding. Last check: 2007-10-24)
		Perpendicular bisectors are erected at the midpoints M
		The case of altitudes reduces to the case of perpendicular bisectors with the following trick.
		The easiest way to see that the radical axis is perpendicular to the center line is to choose the coordinates so as to make the centers lie on the x-axis.
	www.cut-the-knot.com /triangle/remarkable.shtml (1236 words)

	SparkNotes: Constructions: Dividing Angles and Segments
		With angle bisectors and trisectors, it also holds true that any of the new angles created by the bisector or trisector is equal to exactly one-half or one-third of the original angle, depending on whether the angle has been bisected or trisected.
		If a bisector is perpendicular to the segment it bisects, it is called the perpendicular bisector of that segment.
		Because there exists only one line perpendicular to a line at a given point, a segment has only one perpendicular bisector: the perpendicular line that passes through the midpoint of the segment.
	www.sparknotes.com /math/geometry1/constructions/section5.rhtml (445 words)

	Centers of a Triangle
		Theorem: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices
		Prove that the three perpendicular bisectors of the sides of a triangle are concurrent.
		Since m is the perpendicular bisector of segment AC, DA =DC by the perpendicular bisector theorem.
	jwilson.coe.uga.edu /EMT668/EMAT6680.2003.Su/Glynn/4690A04.htm (301 words)

	Formal Constructions: Part 1
		Bisector of a line segment: constructing a bisector of a line segment.
		This procedure is also used to construct a perpendicular bisector.
		Perpendicular lines: constructing a perpendicular line to a given line from a point on the line and constructing a perpendicular line to a given line from a point not on a line.
	www.teacherlink.org /content/math/interactive/geoweb/lessons/construction1/home.html (229 words)

	TopSmart.. Not just Smart... TopSmart
		Angle bisector: If a ray contains the vertex of a given angle and divides the angle into two equal angles, then that ray is an angle bisector.
		Perpendicular bisector: A line perpendicular to the midpoint of a segment.
		A point is on a perpendicular bisector, if and only if, it is equidistant from the endpoints of the line segment.
	www.topsmart.com /MathDefine.html (2123 words)

	Math 572, August 30 (Site not responding. Last check: 2007-10-24)
		It follows that C is on the perpendicular bisector of AB.
		The purpose of the last two theorems is to justify saying that the perpendicular bisector is the locus of points equidistant from the endpoints of a segment.
		To do this we need to know that every point that is equidistant is on the perpendicular bisector and every point that is on the perpendicular bisector is equidistant, which is exactly the content of the last two theorems.
	www.math.ksu.edu /math572/notes/902.html (425 words)

	Perpendicular Bisector - interactive applet
		The definition and properties of the perpendicular bisector are explored using a geometry applet.
		The perpendicular bisector of a line segment AB is a line that is perpendicular to AB and passes through the midpoint of segment AB.(see figure below).
		A large screen applet is used to explore the definition and properties of the perpendicular bisector.
	www.analyzemath.com /Geometry/PerpendicularBisector/PerpendicularBisector.html (296 words)

	Concepts to know for Exam II
		Understand why the perpendicular bisector of a segment is used to find the circumcenter (center of a circle circumscribed about a figure) of a circumscribed circle.
		Understand why the angle bisector is used to find the incenter (center of a circle inscribed in a figure) of the inscribed circle.
		Understand why a perpendicular to a segment through the incenter is used to find the length of the radius of the inscribed circle.
	www.math.tamu.edu /~lynnette.cardenas/366/rev2.html (704 words)

	CSUSB Math 129: Constructing perpendiculars
		Definition: The perpendicular bisector of a segment is the line that is perpendicular (at a right angle) to the segment and goes through the midpoint of the segment.
		Theorem: The perpendicular bisector of a segment is the set of all points that are the same distance from both the endpoints of the segment.
		The midpoint of the segment is where the perpendicular intersects the line (by definition).
	www.math.csusb.edu /courses/m129/perp.html (801 words)

	Basic Theorems (Site not responding. Last check: 2007-10-24)
		Two lines M and N are perpendicular if they intersect at a point P and for some points Q on M and R on N, with Q, R distinct from P, a(QPR) = 90.
		The perpendicular bisector of a line segment AB is the line perpendicular to AB through the midpoint of AB.
		Perpendicular bisector: The perpendicular bisector of a line segment AB is the set of all points which are equidistant from A and B. Existence of perpendicular: For every line L and every point A there exists a line through A perpendicular to L. Set 2: Parallels
	www.math.uga.edu /~clint/2004/5200/axioms/theorems.htm (790 words)

	Keymath.com : Discovering Geometry : Dynamic Geometry Explorations : Chord Properties
		The lengths of the segments of chord AB and the distances from the center of the circle to the chords are given.
		Formulate the Perpendicular to a Chord Conjecture: The perpendicular from the center of a circle to a chord is the _____ of the chord.
		Formulate the Perpendicular Bisector of a Chord Conjecture: The perpendicular bisector of a chord _____.
	www.keymath.com /DG/dynamic/chords.html (538 words)

	Angles and Lines
		If a line is perpendicular to a line that is parallel to another, then the first line is perpendicular to the other two, again if they are all coplanar.
		Perpendicular bisector of a segment allowing access on one side only.
		To construct a segment perpendicular bisector, set the compass to a radius maybe 50% longer than the segment, place the point of your compass in turn on each end and draw an arc on each side of the segment such that these arcs intersect.
	www.andrews.edu /~calkins/math/webtexts/geom03.htm (2908 words)

	Perpendicular Bisector
		The perpendicular bisector is a line that is perpendicular to a segment and divides it into two congruent segments.
		This is the perpendicular bisector of segment AB.
		Use the Measure button to verify that line PQ is perpendicular to segment AB and that point M is the midpoint of segment AB.
	illuminations.nctm.org /tools/IGD_lines/PerpendicularBisector.html (489 words)

	Geometry Session 7: Symmetry
		If point A' is the mirror image of point A in a figure with a line of symmetry, then the line of symmetry is the perpendicular bisector of the segment AA'.
		For each figure, reflect the figure over the line shown using perpendicular bisectors.
		In this video segment, the participants explain different ways in which they came up with perpendicular bisectors to reflect the figures over the line of symmetry.
	www.learner.org /channel/courses/learningmath/geometry/session7/part_a/bisector.html (246 words)

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