Summary of mathematics knowledge points in the first volume of the ninth grade.
Definition of 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 passing through 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.
9, medium, A(x 1, y 1), B(x2, y2).
10, tangent judgment of circle.
When (1)d=r, the straight line is the tangent of the circle.
The tangent point is not clear: draw a vertical line to prove the radius.
(2) The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.
Clear tangent point: uniform radius and vertical.
1 1, the properties of the tangent of the circle (supplementary).
(1) The diameter passing through the tangent point must be perpendicular to the tangent.
(2) A straight line passing through the tangent point and perpendicular to the tangent line must pass through the center of the circle.
12, tangent length theorem.
(1) Tangent length: two tangents from a point outside the circle to the circle. The length of the connecting line between the tangent point and the point is called the tangent length of the point to the circle.
(2) Tangent length theorem.
PA and PB cut o at point a and point B.
∴PA=PB,∠ 1=∠2。
13, inscribed circle and related calculation.
(1) The center of the inscribed circle is the intersection of three bisectors of the inner angle, and the distances to the three sides are equal.
(2) As shown in the figure, in △ABC, AB=5, BC=6, AC=7, and ⊙O tangent △ABC is at points D, E and F.
Q: the length of AD, BE and cf.
Analysis: Let AD=x, then AD=AF=x, BD=BE=5-x, CE = CF = 7-X. 。
The equation can be obtained: 5-x+7-x=6, and the solution is x=3.
(3) At △ AB=c, ∠ C = 90, AC=b, BC=a, AB=c.
Find the radius r of inscribed circle.
Analysis: Square ODCE was first proved,
Get CD=CE=r
AD=AF=b-r，BE=BF=a-r
b-r+a-r=c
14, (1) Chord tangent angle: the vertex of the angle is on the circumference, one side of the angle is the tangent of the circle, and the other side is the chord of the circle.
BC cuts ⊙O at point B, AB is the chord, ∠ABC is the chord tangent angle, ∠ ABC = ∠ D.
(2) Intersecting chord theorem.
The two chords AB and CD of a circle intersect at point p, so PA? PB=PC？ PD .
(3) Cutting line theorem.
As shown in the figure, PA secant ⊙O at point A and PBC secant ⊙O, so PA2=PB? Personal computer.
(4) Inference: As shown in the figure, if PAB and PCD are secant of ⊙O, then PA? PB=PC？ PD .
15, the positional relationship between circles.
(1) external: D >；; R 1+r2, with 0 intersections;
Circumscribed: d=r 1+r2, with 1 intersections;
Intersection point: r 1-r2
Signature: d=r 1-r2, with 1 intersections;
Include: 0≤d
(2) nature.
The intersection of two circles bisects the common chord vertically.
The straight line connecting two circles must pass through the tangent point.
16, calculation of correlation quantity in circle.
(1) The arc length is represented by L, the central angle is represented by N, and the radius of the circle is represented by R. ..
(2) The area of the sector is represented by S. ..
(3) The lateral development of the cone is fan-shaped.
R is the radius of the bottom circle, and A is the length of the bus.
The first volume of ninth grade mathematics knowledge points induction II
1 quadratic root: the formal formula is quadratic root;
Attribute: It is non-negative;
2 multiplication and division of quadratic root:
Quadratic root addition and subtraction: when adding and subtracting quadratic roots, first merge the simplest quadratic roots of Huawei, and then merge the quadratic roots with the same number of roots.
4 Helen-Qin Jiushao formula: where s is the area and p is.
1: An equation with algebraic expressions on both sides of the equal sign and only one unknown number. The number of unknowns is 2.
2 matching method: one side of the equation is matched in a completely flat way, and then the two sides are squared;
Factorization: the product of two factors on the left and zero on the right.
3 The application of quadratic equation in one variable in practical problems
Vieta Theorem: Let it be the two roots of the equation, then there is.
1: graphic transformation in which a graphic rotates by an angle around a certain point.
Property: the distance from the corresponding point to the center is equal;
The included angle between the corresponding point and the line segment connecting the rotation center is equal to the rotation angle.
Graphic consistency before and after rotation.
2. Center symmetry: if one graph rotates around a point by 180 degrees and coincides with another graph, then the two graphs are symmetrical about the center of this point;
Centrally symmetric figure: a figure rotates 180 degrees around a certain point and can coincide with the original figure, so it is called centrosymmetric;
Three coordinates of a point with symmetrical origin.
1 Definition of circle, center, radius, diameter, arc, chord and semicircle
2 Diameter perpendicular to the chord
A circle is a figure, and any straight line with a diameter is its axis of symmetry;
The diameter perpendicular to the chord divides the chord in two and squares the two arcs opposite to the chord;
The diameter of the chord is perpendicular to the chord, and the two arcs opposite the chord are equally divided.
3 Arc, chord and central angle
In the same circle or in the same circle, equal central angles have equal arcs and equal chords.
4 circle angle
In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half the central angle of the arc;
The semicircle (or diameter) faces the right angle, and the 90-degree angle faces the diameter.
The positional relationship between five points and a circle
D>r, the point outside the circle
Points on the circle d=r
Point dR+r in the circle
Circumscribed =R+r
Intersecting R-r
The ninth grade, the first volume, mathematics knowledge points induction III
Parabolic vertex coordinate formula
y=ax2+bx+c(a=？ 0) Yes (-b/2a, (4ac-b2)/4a).
The vertex coordinates of y=ax2+bx are (-b/2a, -b2/4a).
Relevant conclusions
Parabola y 2 = 2px (p > 0) The focus F is a straight line L with an inclination angle of θ, and L intersects the parabola at A(x 1, y 1), B(x2, y2), and
① x 1x2 = p 2/4, y 1y2 =-p 2, which can only be established when the straight line passes through the focus;
② chord length of focus: | ab | = x1+x2+p = 2p/[(sinθ) 2];
③( 1/| FA |)+( 1/| FB |)= 2/P；
④ If OA is perpendicular to OB, AB crosses the fixed point M(2P, 0);
⑤ Focus radius: |FP|=x+p/2 (the distance from a point P on the parabola to the focus F is equal to the distance from the directrix L);
⑥ chord length formula: ab = √ (1+k2) │ x2-x1│;
⑦△=b^2-4ac；
(8) The vertical distance from the focus of a parabola to its tangent is the median of the distance from the focus to the tangent point and the distance from the vertex;
Pet-name ruby The standard form of the tangent of parabola at x0 and y0 is yy0=p(x+x0).
⑴△=b^2-4ac>； 0 has two real roots;
⑵ = b 2-4ac = 0 has two identical real roots;
⑶△=b^2-4ac<； 0 has no real root.
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