# Vertical line

## defined

through the midpoint of a line segment, and a line perpendicular to this segment, called perpendicular bisector of this segment , also known as " the vertical. "

in FIG. 1, N is the midpoint of AB, through the N-point for MN⊥AB, then, MN is the perpendicular bisector of the line AB.

## properties

(1) perpendicular to the perpendicular bisector of the line segment bisects it

(2) perpendicular bisector of any point, the end points of the line segment equidistant

(3) three sides of a triangle perpendicular bisectors intersect at a point, called the circumcenter point, and from this point to the three vertices are equal

(4) < b> perpendicular bisectors determined : must satisfy (1) through the straight line midpoint; (2) a straight line segment ⊥

h2> Inverse Theorem

inverse theorem: a line segment equal to the distance of two end points, perpendicular bisector of the line segment in this.

1, N is the midpoint of AB is known, the MN is the perpendicular bisector of AB, on a plane P that satisfies PA = PB, demonstrated: in the P MN.

Solution:

∵MN is the perpendicular bisector of AB

∴AN = BN

∵PA = PB, PN = PN

∴ △ PAN≌ △ PBN

∴∠PNA = ∠PNB

∵∠PNA + ∠PNB = 180 °

∴∠ PNA = ∠PNB = 90 °

on the plane due to over-point, and only a straight line perpendicular to the vertical is known, it is on the P MN.

This inverse theorem proved.

## The method of determining

① using defined: After the midpoint of a line segment, and a line perpendicular to this line is the perpendicular bisector of the line segment

② an equal distance to the line segment end points two points, the perpendicular bisector of this segment. (I.e. the perpendicular bisector of the line segment can be regarded as equal to the set point distance segment end points).

## mapping method

(1) Method Ruler

a., Respectively to two ends of the lines as the center, to is greater than line segment one-half the length of is the radius of the arc drawn, to give two intersection points (cross on both sides of the intersection of two segments)

b. connecting the two intersection points

(2) Measuring method

(3) origami method (folding process)

## axis of symmetry

when the pattern (the pattern may be linear, polyline , curvilinear) symmetrical about a straight line, this axis is called the axis of symmetry. In an example pentagram, which has five axis of symmetry.

is the perpendicular bisector of a line segment when there is the presence of this concept. It is defined through the midpoint of a line segment, and a line perpendicular to this segment, this segment is called the perpendicular bisector (the vertical). It has some limitations.

symmetry axis symmetry axis of symmetry is any perpendicular bisector of two points corresponding to the connection section.

## Life Application

have A, B, C (not on the same straight line) three village, is now ready to build a school, to school requires three equidistant from villages Please determine the location of the school.

Analysis: sequentially connected AB, AC, BC, as AB, BC perpendicular bisector intersect at point O, by the nature of the perpendicular bisectors have OA = OB, OB = OC, so that OA = OB = OC, O is the location of the school.

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