Knowledge points of mathematics unit in the first volume of the ninth grade
Chapter I Evidence
I. Isosceles triangle
1. Definition: A triangle with two equal sides is an isosceles triangle.
2. Property: 1. The two base angles of an isosceles triangle are equal (abbreviated as "equilateral angles").
2. The bisector of the top angle of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide ("three lines are one")
3. The bisectors of the two base angles of an isosceles triangle are equal. (The midline of the two waists is equal and the height of the two waists is equal)
4. The point on the vertical line of the bottom of the isosceles triangle is equal to the distance between the two waists.
5. The included angle between waist height and waist bottom of isosceles triangle is equal to half of the top angle.
6. The sum of the distances from any point on the bottom of an isosceles triangle to two waists is equal to the height of one waist (which can be proved by equal area method).
7. An isosceles triangle is an axisymmetric figure with only one axis of symmetry, and the line where the bisector of the top angle is located is its axis of symmetry.
3. Judgment: In the same triangle, two triangles with equal angles are isosceles triangles (equilateral triangles for short).
Special isosceles triangle
equilateral triangle
1, definition: A triangle with three equilateral sides is called an equilateral triangle, also called a regular triangle.
(Note: If all three sides of a triangle are equal, it is called an equilateral triangle, but generally it is not called an isosceles triangle).
2. Properties: (1) The internal angles of equilateral triangles are all equal, which are all 60 degrees.
(2) The midline, the high line of each side of the equilateral triangle and the bisector of each corner coincide with each other.
(3) An equilateral triangle is an axisymmetric figure with three axes of symmetry, and the axis of symmetry is the straight line where the median line, height line or bisector of each side is located.
3. Judgment: (1) A triangle with three equal sides is an equilateral triangle.
(2) Three triangles with equal internal angles are equilateral triangles.
(3) An isosceles triangle with an angle of 60 degrees is an equilateral triangle.
(4) A triangle with two angles equal to 60 degrees is an equilateral triangle.
Summary of mathematics knowledge points in the second volume of the ninth grade
The positional relationship between straight line and circle
(1) A straight line and a circle have nothing in common, which is called separation. AB is separated from circle O, d>r.
② A straight line and a circle have two common points, which are called intersections. This straight line is called the secant of a circle. AB intersects with ⊙O and d.
③ A straight line and a circle have only one common point, which is called tangency. This straight line is called the tangent of the circle, and this common point is called the tangent point. AB is tangent to ⊙O, and d = r. (d is the distance from the center of the circle to the straight line)
In the plane, the general method to judge the positional relationship between the straight line Ax+By+C=0 and the circle X 2+Y 2+DX+EY+F = 0 is:
1. You can get y=(-C-Ax)/B from Ax+By+C=0 (where b is not equal to 0), and substitute it into x 2+y 2+dx+ey+f = 0, and the equation about x becomes.
If b 2-4ac > 0, the circle and the straight line have two intersections, that is, the circle and the straight line intersect.
If b 2-4ac = 0, the circle and the straight line have 1 intersections, that is, the circle is tangent to the straight line.
If b 2-4ac
2. If B=0 indicates that the straight line is Ax+C=0, that is, x=-C/A, parallel to the Y axis (or perpendicular to the X axis), change X 2+Y 2+DX+EY+F = 0 to (X-A) 2+(Y-B) 2 = R, and let Y =
When x=-C/Ax2, the straight line deviates from the circle;
Rotational transformation
1. Concept: In a plane, turning a figure around a fixed point by an angle in a certain direction is called rotation.
Description: (1) The rotation of the graph is determined by the rotation center and rotation angle; (2) In the process of rotation, the center of rotation remains unchanged. (3) In the process of rotation, the rotation direction is consistent. (4) When the rotation is stationary, the rotation angle of a point on the diagram is the same. (5) Rotation will not change the size and shape of the figure.
2. Property: (1) The distance from the corresponding point to the rotation center is equal;
(2) The included angle of the connecting line between the corresponding point and the rotation center is equal to the rotation angle;
(3) Graphic congruence before and after rotation.
3. Steps and methods of rotation drawing: (1) Determine the rotation center, rotation direction and rotation angle; (2) Find out the key points of the graph; (3) connecting the key points of the graph with the rotation center, and then rotating by a rotation angle according to the rotation direction to obtain the corresponding points of these key points; (4) Connect these corresponding points in turn according to the original image, and the resulting graph is the rotated graph.
Note: When drawing by rotation, the included angle between a pair of corresponding points and the rotation center is the rotation angle.
The ninth grade mathematics review materials volume one
Knowledge point 1: the basic concept of unary quadratic equation
1, the constant term of the unary quadratic equation 3x2+5x-2=0 is -2.
2. The coefficient of the primary term of the unary quadratic equation 3x2+4x-2=0 is 4, and the constant term is -2.
3. The quadratic term coefficient of the unary quadratic equation 3x2-5x-7=0 is 3, and the constant term is -7.
4. Transform the equation 3x(x- 1)-2=-4x into the general formula 3x2-x-2=0.
Knowledge point 2: Cartesian coordinate system and the position of points
1. In the rectangular coordinate system, point A (3 3,0) is on the Y axis.
2. In the rectangular coordinate system, the abscissa of any point on the X axis is 0.
3. In rectangular coordinate system, point A (1, 1) is in the first quadrant.
4. In rectangular coordinate system, point A (-2,3) is in the fourth quadrant.
5. In rectangular coordinate system, point A (-2, 1) is in the second quadrant.
Knowledge point 3: Find the function value of the known independent variable.
1. When x=2, the value of function y= is 1.
2. When x=3, the value of function y= is 1.
3. When x=- 1, the value of function y= is 1.
Knowledge point 4: the concept and nature of basic functions
1 and function y=-8x are linear functions.
2. The function y=4x+ 1 is a proportional function.
3. This function is an inverse proportional function.
4. The opening of parabola y=-3(x-2)2-5 is downward.
5. The symmetry axis of parabola y=4(x-3)2- 10 is x=3.
6. The vertex coordinate of parabola is (1, 2).
7. The image of the inverse proportional function is in the first and third quadrants.
Knowledge point 5: mean, median and mode of data
1, data 13, 10,12,8,7, the average value is 10.
2. The pattern of data 3, 4, 2, 4, 4 is 4.
3. The median of data 1, 2,3,4,5 is 3.
Su Ke version of the third grade mathematics knowledge points to sort out related articles;
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★ sort out and summarize the knowledge points of mathematics in grade three.
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★ Knowledge points of the first volume of mathematics in the third grade
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