Mathematics knowledge points in the new semester of grade three
First, the definition of a circle
1, a graph composed of points with a fixed point as the center and a fixed length as the radius.
2. A graph composed of points with the same distance to a fixed point on the same plane.
Second, the elements of a circle
1, radius: the line connecting a point on a circle with the center of the circle.
2. Diameter: Two points on the connecting circle have a line segment passing through the center of the circle.
3. Chord: a line segment connecting two points on a circle (the diameter is also a chord).
4. Arc: The curve between two points on a circle. A semicircle is also an arc.
(1) Bad arc: the arc is less than half a circle.
(2) Optimal arc: an arc larger than half a circle.
5. Central angle: an edge with the center of the circle as the vertex and the radius as the angle.
6. Circumferential angle: the vertex is on the circumference, and the two sides of the circumferential angle are chords.
7. Chord center distance: the length from the center of the chord to the vertical section of the chord.
Third, the basic properties of the circle
1, symmetry of circle
(1) A circle is a figure, and its symmetry axis is the straight line where the diameter lies.
(2) A circle is a figure with a symmetrical center, and its symmetrical center is the center of the circle.
(3) The circle is a symmetrical figure.
2. Vertical diameter theorem.
(1) bisects the chord perpendicular to its diameter and bisects the two arcs opposite the chord.
(2) Inference:
Bisect the diameter (non-diameter) of a chord, perpendicular to the chord and bisecting the two arcs opposite the chord.
Bisect the diameter of the arc and bisect the chord of the arc vertically.
3. The degree of the central angle is equal to the degree of the arc it faces. The degree of the circle angle is equal to half the radian it subtends.
(1) The circumferential angles of the same arc are equal.
(2) The circumferential angle of the diameter is a right angle; The angle of a circle is a right angle, and the chord it subtends is a diameter.
4. In the same circle or equal circle, as long as one of the five pairs of quantities, namely two chords, two arcs, two circumferential angles, two central angles and the distance between the centers of two chords, is equal, the other four pairs are also equal.
5. The two arcs sandwiched between parallel lines are equal.
6. Let the radius of ⊙O be r and op = d. ..
7.( 1) The center of the circle crossing two points must be on the vertical line connecting the two points.
(2) Three points that are not on the same straight line determine a circle, the center of which is the intersection of the perpendicular lines of three sides, and the distances from this point to these three points are equal.
The outer center of the right angle is the midpoint of the hypotenuse. )
8. The positional relationship between a straight line and a circle. D represents the distance from the center of the circle to a straight line, and r represents the radius of the circle.
A straight line and a circle have two intersections, and the straight line and the circle intersect; There is only one intersection point between a straight line and a circle, and the straight line is tangent to the circle;
There is no intersection between a straight line and a circle, but a straight line and a circle are separated.
Nine-grade mathematics knowledge points
Essential knowledge points of circle
(1) circle
In a plane, a closed curve formed by a moving point rotating around a certain point with a certain length is called a circle. A circle has countless axes of symmetry.
(2) Related characteristics of the circle
1) diameter
The line segment connecting the center of the circle and any point on the circle is called radius, and the letter is expressed as R.
The line segment passing through the center of the circle with both ends on the circle is called the diameter, and the letter is denoted as d.
A straight line with a diameter is the symmetry axis of a circle. In the same circle, the diameter of the circle is d=2r.
2) Chords
A line segment connecting any two points on a circle is called a chord. The longest chord in the same circle is the diameter. The straight line with the diameter is the symmetry axis of the circle, so there are countless symmetry axes of the circle.
3) Arc
The part between any two points on a circle is called an arc, which is represented by "⌒".
An arc larger than a semicircle is called an optimal arc, and an arc smaller than a semicircle is called a suboptimal arc, so a semicircle is neither an optimal arc nor a suboptimal arc. The optimal arc is generally represented by three letters, and the suboptimal arc is generally represented by two letters. The optimal arc is an arc with a central angle greater than 180 degrees, and the suboptimal arc is an arc with a central angle less than 180 degrees.
In the same circle or equal circle, two arcs that can overlap each other are called equal arcs.
4) Angle
The angle of the vertex on the center of the circle is called the central angle.
The angle at which the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle. The angle of a circle is equal to half the central angle of the same arc.
Knowledge points of mathematics review materials for grade three
Axisymmetric knowledge point
1. If a graph is folded along a straight line and the parts on both sides of the straight line can overlap each other, then the graph is called an axisymmetric graph; This straight line is called the axis of symmetry.
2. The symmetry axis of an axisymmetric figure is the perpendicular bisector of a line segment connected by any pair of corresponding points.
3. The distance from the point on the bisector of the angle is equal to both sides of the angle.
4. The distance between any point on the vertical line of the line segment and the two endpoints of the line segment is equal.
5. The point with equal distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
6. The corresponding line segment and the corresponding angle on the axisymmetric figure are equal.
7. Draw an axisymmetric figure about a straight line: find the key points, draw the corresponding points of the key points, and connect the points in the original order.
8. The coordinates of the point (x, y) about the axis symmetry of X are (x, -y).
The coordinates of the point (x, y) that is symmetric about y are (-x, y).
The coordinates of the point (x, y) that is symmetrical about the origin are (-x, -y).
9. The nature of isosceles triangle: the two base angles of isosceles triangle are equal (equilateral and equiangular).
The bisector of the top angle of an isosceles triangle, the height on the bottom edge and the midline on the bottom edge coincide, which is called the integration of the three lines for short.
10. Determination of isosceles triangle: equilateral and equilateral.
1 1. The three internal angles of an equilateral triangle are equal and equal to 60.
12. Determination of equilateral triangle: A triangle with three equal angles is an isosceles triangle.
An isosceles triangle with an angle of 60 is an equilateral triangle.
A triangle with two angles of 60 is an equilateral triangle.
13. In a right triangle, the right angle side of 30 is equal to half of the hypotenuse.
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