A and B in the formula (1) can be monomials or polynomials.
(2) If the formula cannot be directly applied, we should be good at transforming and applying the formula.
(a), change the symbol
Example: Use the complete square formula to calculate:
( 1)(-4x+3y)2
(2)2
Analysis: This example changes the signs of A and B in the formula. Taking the second subtitle as an example, the simplest way to deal with this problem is to regard (-a) in this formula as A in the original formula and (-b) as B in the original formula, and then directly apply the formula to calculate.
Answer:
( 1) 16x2—24xy+9y2
(2)a2+2ab+b2
(2) Number of variables:
Example: Calculation: (3a+2b+c)2
Analysis: the left side of the complete square formula is the multiplication of two identical binomials, but there are three items in this example, and two of them should be considered as one item to solve the contradiction. Therefore, when using the formula, (3a+2b+c)2 can be converted into [(3a+2b)+c]2, and the formula can be directly applied for calculation.
Answer: 9a2+ 12ab+6ac+4b2+4bc+c2.
(3) Variable structure
Example: Calculate by formula:
( 1)(x+y)(2x+2y)
(2)(a+b)(—a—b)
(3)(a-b)(b-a)
Analysis; In this example, all binomials are multiplied by binomials. On the surface, the appearance structure does not conform to the characteristics of the formula, but it is easy to find that one of the factors can be properly deformed, that is,
( 1)(x+y)(2x+2y)=2(x+y)2
(2) (a+b)(—a—b)=—(a+b)2
(3)(a-b)(b-a)=-(a-b)2
The first volume of eighth grade mathematics, knowledge points of mathematical formulas, 2, congruent triangles
1, definition: Two triangles that can completely coincide are called congruent triangles.
Understand:
(1) the shape and size of congruent triangles are completely equal, regardless of location;
② A triangle can be translated, folded and rotated to get its congruence;
③ The congruence of triangle does not change with the change of position.
2. What is the nature of congruent triangles?
(1) congruent triangles has equal sides and angles.
Understand:
① Long side to long side, short side to short side; Maximum angle to maximum angle and minimum angle to minimum angle;
② The opposite side of the corresponding angle is the corresponding edge, and the angle of the corresponding edge pair is the corresponding angle.
(2) The circumference and area of congruent triangles are equal.
(3) The corresponding median line, angular bisector and high line on the corresponding side of congruent triangles are equal respectively.
3. congruent triangles's judgment
Edge: Three edges correspond to the coincidence of two triangles (abbreviated as "SSS")
Angle: Two sides and their included angles are equal. Two triangles are congruent (abbreviated as "SAS").
Corner: Two triangles coincide with two corners and their clamping edges (abbreviated as "ASA").
Corner edge: the opposite side of two angles and one angle corresponds to the congruence of two triangles (abbreviated as "AAS")
Hypotenuse and right-angled side: hypotenuse and a right-angled side correspond to the congruence of two right-angled triangles (abbreviated as "HL").
Second, the bisector of the angle:
A ray drawn from the vertex of an angle divides the angle into two equal angles. This ray is called the bisector of the angle.
1, property: the distance from the point on the bisector of the angle is equal to both sides of the angle,
2. Judgment: The point with equal distance from the inside of the corner to both sides of the corner is on the bisector of the corner.
Three, learning congruent triangles should pay attention to the following questions:
(1) The different meanings of "corresponding edge" and "opposite edge", "corresponding angle" and "diagonal" should be correctly distinguished;
(2) When two triangles are congruent, the letters representing the corresponding vertices should be written in the corresponding positions;
(3) Two triangles with "three corresponding angles are equal" or "two opposite corners have two sides and one of them is equal" are not necessarily the same;
(4) Always pay attention to the implicit conditions in graphics, such as "corner", "edge" and "diagonal".
(5) Prove triangle congruence by truncated complementary method.
The first volume of eighth grade mathematics, mathematical formula knowledge points, 3, axisymmetric graphics
1. Fold the chart along a straight line. If the parts on both sides of a straight line can completely overlap, then this graph is called an axisymmetric graph. This straight line is its axis of symmetry. At this time, we also say that this figure is symmetrical about this straight line (axis).
2. Fold the chart along a straight line. If it can completely coincide with another figure, the two figures are said to be symmetrical about this line. This straight line is called the axis of symmetry. The point that overlaps after folding is the corresponding point, which is called the symmetrical point.
3. The difference and connection between axisymmetric figure and axisymmetric figure.
4. Properties of Axisymmetric and Axisymmetric Graphs
① Two figures symmetrical about a straight line are conformal.
(2) If two figures are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.
③ The symmetry axis of an axisymmetric figure is the median vertical line of a line segment connected by any pair of corresponding points.
(4) If the straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.
⑤ Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
Second, the vertical line of the line segment
1, definition: A straight line passing through the midpoint of a line segment and perpendicular to this line segment is called the median line of this line segment, also called the median line.
2. Property: The distance between the point on the vertical line of the line segment and the two endpoints of the line segment is equal.
3. Judgment: The point where the distance between the two ends of a line segment is equal is on the middle vertical line of the line segment.
Third, use coordinates to express the axisymmetric summary:
In a plane rectangular coordinate system
(1) The abscissas of the points symmetrical about the X axis are equal, and the ordinate is reciprocal;
(2) The abscissas of the points symmetrical about the Y axis are opposite to each other, and the ordinate is equal;
③ The abscissa and ordinate of a point symmetrical about the origin are opposite numbers;
(4) the horizontal (vertical) coordinate relationship between two points on a straight line parallel to the X axis or the Y axis;
⑤ About the coordinates symmetrical to the straight line X=C or y = c.
The coordinates of the point (x, y) on the axis symmetry of X are _ (x, -y)_ _ _ _ _ _ _,
The coordinates of the point (x, y) symmetric about y are _ _(-x, y)_ _ _ _ _,