Mathematics in the second day of junior high school-Axisymmetry-The perpendicular bisector of a line segment; The two ends of the line segment are taken as the center, and half the length of the line segment is taken as the radius to arc up and down to get two intersections.
The two points of the connecting line are line segments.
The nature of the midline is that the distance from any point on the midline of a line segment to both ends of the line segment is equal.
The Properties of the Median Vertical Line and the Properties of the Median Vertical Line
1. perpendicular bisector is vertical, divide its upper line equally.
2. The distance from any point on the perpendicular bisector to both ends of the line segment is equal.
Inverse theorem: the point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.
Prove the nature of the median vertical. Hello:
A straight line perpendicular to and bisecting a line segment is the median vertical line of this line segment.
Has the following important attributes:
The point on the vertical line of a line segment is equal to the distance between the two endpoints of the line segment.
Prove:
Let the middle vertical line of line segment AB be PQ and intersect with line segment AB at point P, then
It is proved that QA=QB,
According to the definition of the median vertical line, I know
PA=PB
Then draw a conclusion from Pythagorean theorem.
QA？ =PA？ +PQ？ ，QB？ =PB？ +PQ？ ,…①
PA = PB
∴PA？ =PB？ ,…②
PQ again? =PQ？ ,…③
∴ Combined with ① ② ③ formula, we can get
QA？ =QB？
QA > 0，QB>0，
∴QA=QB
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What is the nature and significance of the median vertical line? Perpendicular bisector:
1. Theorem: The distance between a point on the vertical line of a line segment and its two endpoints is equal.
Inverse theorem: the point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.
2. Another definition of the midline of a line segment: the midline of a line segment can be regarded as a set of all points with equal distances from the two endpoints of the line segment.
Perpendicular bisector definition
A straight line passing through the midpoint of a line segment and perpendicular to the line segment is called the median vertical line (vertical line) of the line segment. The median vertical line, referred to as "median vertical line", is a very important part of junior high school geometry.
Properties of straight bisector
1. perpendicular bisector is vertical, divide its upper line equally. 2. The distance from any point on the perpendicular bisector to both ends of the line segment is equal. 3. The perpendicular lines of the three sides of a triangle intersect at a point, which is called the outer center, and the distance from the point to the three vertices is equal (and the distance is the shortest, only this one).
Perpendicular bisector inverse theorem
The point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment. Systematic combination
As shown in the figure: the straight line MN is the middle vertical line of the line segment AB. Note: To prove that a straight line is the middle perpendicular of a line segment, it is necessary to prove that the distance between two points and the line segment is equal and both points are on the straight line to prove that in general, the middle perpendicular and congruent triangles are used. The nature of the median vertical line: the point on the median vertical line of a line segment is equal to the distance between the two endpoints of the line segment. Clever method: the distance from the point to both ends of the line segment is equal. Congruent triangles can be used to prove it.
Perpendicular bisector ruler exercise
One method: (plan with compasses) 1. Find the center of the line segment so that the vertical segment of the line segment passes through this point. 2. Draw an arc with the two endpoints of the line segment as the center and the radius greater than half the length of the line segment. Find the intersection point (two intersections intersect with the same side of the line segment). 3. Connect these two intersections. Principle: The height of an isosceles triangle bisects the base vertically. Method 2: 1, connecting these two intersections. Principle: two points are on a line. The nature of isosceles triangle: 1, three lines in one (the high line of the bottom, the middle line of the bottom and the bisector of the top corner of isosceles triangle coincide. ) 2, equilateral 3, equilateral angle
Property theorem of median vertical line 1. Perpendicular bisector is vertical and bisects the upper part of its straight line.
2. The distance from any point on the perpendicular bisector to both ends of the line segment is equal.
3. The perpendicular lines of the three sides of a triangle intersect at a point, which is called the outer center, and the distance from the point to the three vertices is equal.
As long as the bisector of the angle is 1 1.3, then the angle is an axisymmetric figure, and the straight line where the bisector of the angle lies is its axis of symmetry. The point on the bisector of an angle is equal to the distance on both sides of the angle.
On the bisector of an angle, a point with equal distance from the inside of the angle to both sides of the angle.
Mathematical properties: angular bisector and angular bisector of perpendicular bisector;
1. Theorem 1: The distance from a point on the bisector of an angle to both sides of the angle is equal;
Theorem 2: Equidistant points on both sides of an angle are on the bisector of this angle.
2. Another definition of the bisector of an angle: the bisector of an angle is the set of all points with the same distance to both sides of the angle.
Perpendicular bisector:
1. Theorem: The distance between a point on the vertical line of a line segment and its two endpoints is equal.
Inverse theorem: the point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.
2. Another definition of the midline of a line segment: the midline of a line segment can be regarded as a set of all points with equal distances from the two endpoints of the line segment.
What is the nature of the median vertical line?
Theorem of properties of angular bisector: a point on the angular bisector is equal to the distance on both sides of the angle.
The nature of parallelogram and its judgment: the opposite sides are parallel and equal, the diagonal lines are equally divided, and the diagonal lines are equal.
The nature of rectangle and its judgment: nature: all four corners are right angles and diagonal lines are equal.
Judgment: A quadrilateral with three right angles is a rectangle. Parallelograms with equal diagonal lines are rectangles,
The nature of diamond and its judgment: four sides are equal; Diagonal lines are bisected vertically, and each diagonal line bisects a set of diagonal lines.
Judgment: a quadrilateral with four equal sides is a diamond;
Parallelograms with diagonal lines perpendicular to each other are diamonds.
The nature of a square and its judgment: four sides are equal; The four angles are right angles; The diagonal line is divided into two, perpendicular and equal to each other.
And each diagonal bisects a set of diagonal lines.
Decision: A diamond with equal diagonal lines is a square.
The rectangle with vertical diagonal is a square.
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