What is the summary of the knowledge points of solving right triangles?

The relationship between the corners of an isosceles right triangle :

1) The sum of the three internal angles of the triangle is equal to 180°;

2) One outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it;

3) One outer angle of the triangle is greater than any of the inner angles that are not adjacent to it;

4) The sum of the two sides of the triangle is greater than the third side, and the difference between the two sides is less than the third side;

5) Within the same triangle, the big side is against the big angle, and the big angle is against the big side.

There are four special line segments in an isosceles right triangle: the angular bisector, the median, the high, and the median.

1) The intersection of the angular bisector of the triangle is called the inner part of the triangle, which is the center of the inscribed circle of the triangle, and its distance to each side is equal.

The circumscribed center of the triangle, the outer center, is the intersection of the perpendicular bisector of the three sides of the triangle, and it is at an equal distance from the three vertices).

2) The intersection of the three midlines of the triangle is called the center of gravity of the triangle, and the distance from each vertex is equal to 2 times its distance from the midpoint of the opposite side.

The summary of the knowledge points of solving right triangles is that the relationship between angles, the mutual surplus of two acute angles, and the relationship between edgesPythagorean theorem, corner relations,Acute trigonometric functions, the basic type of solution of a right triangle and its solution, are knownBeveled edgesand an acute angle to solve a right triangle, a known right angle side and an acute angle to solve a right triangle, known two sides to solve the right triangle, the application of solving the right triangle, the key is to transform the practical problem into a mathematical problem to solve.

Solving a right triangle is a professional term, pinyin is jiězhí jiǎo sān jiǎo xínɡ, in a right triangle, in addition to the right angle, there are five elements in a state, that is, three sides and two acute angles, the process of finding all unknown elements from the known elements in the right triangle except the right angle is called solving the right triangle, and solving the right triangle requires two elements that are purely hand-masked in addition to the right angle, and at least one element is an edge.

The contents of a right triangle

In a right triangle, there are five elements in addition to the right angle, that is, three sides and two acute angles, and the process of finding all the unknown elements in the right triangle except the right angle is called solving the right triangle.

A right triangle is a geometric figure.

It is a triangle with an angle of right angles, and there are ordinary right triangles and isosceles right triangles.

Two kinds, which conform to the Pythagorean theorem, have some special properties and judgment methods.

In a right-angled triangle, the height on the hypotenuse is the mid-term in the proportion of the projection of the two right-angled edges on the hypotenuse.

Each right-angled edge is the mid-scale term between the projection of the right-angled edge on the hypotenuse and the hypotenuse. It is an important theorem for mathematical graph computing.

Summary of knowledge points:

1. Congruent figures, congruent triangles:

1.Congruent Graphs: Two graphs that can coincide exactly are congruent graphs.

2.The properties of congruent figures: the corresponding edges and corresponding angles of congruent polygons are equal.

3.Congruent Triangle:

A triangle is a special polygon, so the corresponding sides and corresponding angles of a congruent triangle are equal. Similarly, if the sides and corners of two triangles correspond to each other, then the two triangles are congruent.

Description: Congruent triangles correspond to the height on the edge, the midline is equal, and the bisector of the corresponding angle is equal; Congruent triangles are equal in circumference and area.

Note: (1) Two triangles with equal circumferences are not necessarily congruent;

2) Two triangles with equal area are not necessarily congruent.

2. Determination of congruent triangles:

1.Determination of the congruence of triangles in general.

1) Edges. Edge Axiom: Three sides correspond to two equal triangle congruences ("edge-edge-edge" or "sss").

2) Corner axioms: two sides and their angles correspond to two triangles that are equally congruent ("corner edges" or "sas").

3) Corner Corner Axiom:

The two corners and their edges correspond to two equal triangle congruence ("corner corners" or "asa").

4) Corner edge theorem: There are two corners and the opposite side of one of the corners corresponds to two equal triangle congruence ("corner edges" or "AAS").

2.Determination of the congruence of right triangles.

The congruence of right triangles can be proved by the determination of the congruence of triangles

An hypotenuse and a right-angled edge correspond to two equal right-angled triangles congruence ("hypotenuse, right-angled" or "hl").

Note: Two triangles with a pair of angles (SSA) and triangles (AAA) on both sides are not necessarily congruent.

3. The nature and determination of the angle bisector:

Property theorem: The distance from a point on the bisector of an angle to both sides of the angle is equal.

Decision theorem: Points with equal distances to both sides of the angle are bisector of the angle.

Fourth, the basic method steps to prove the congruence of two triangles or use it to prove the equality of line segments or angles:

1.Determine known conditions (including implied conditions such as common edges, common angles, vertex angles, angle bisectors, midlines, heights, isosceles triangles, etc.);

2.Review the triangle determination axiom and figure out what else is needed; 3.Write the proof format correctly (the order and correspondence are derived from the known questions to be proved).

1. According to the side: ordinary triangle, isosceles triangle (in an isosceles triangle, a triangle with an equal waist and a bottom is an equilateral triangle.) ）

2. According to the angle: acute triangle, right triangle, obtuse triangle (the two acute angles of the right triangle are redundant. ）

2. The concept of knowledge:

1.Triangle: A figure consisting of three line segments that are not on the same line that meet each other end to end is called a triangle.

2.Trilateral relationship: The sum of any two sides of the triangle is greater than the third side, and the difference between any two sides is less than the third side.

3.Height: A perpendicular line from one of the vertices of the triangle to the line where its opposite edge is located, and the line segment between the vertex and the perpendicular foot is called the height of the triangle.

4.Midline: In a triangle, the segment that connects a vertex to the midpoint of its opposite side is called the midline of the triangle. The three midlines of the triangle intersect at one point, which is called the center of gravity of the triangle.

5.Angle bisector: The bisector of an inner angle of a triangle intersects the opposite side of the angle, and the line segment between the vertices and intersections of this angle is called the angular bisector of the triangle.

6.Stability of the triangle: The shape of the triangle is fixed, and this property of the triangle is called the stability of the triangle.

7.Polygons: In a plane, a shape consisting of a number of line segments that meet each other one after the other is called a polygon.

8.The inside angles of a polygon: The angles formed by the two adjacent sides of a polygon are called its interior angles.

9.The outer corner of the polygon: The angle formed by the extension of one side of the polygon and its adjacent edge is called the outer corner of the polygon.

10.Diagonal of a polygon: A segment of a line that connects two vertices that are not adjacent to a polygon, called the diagonal of a polygon.

11.Regular polygon: A polygon in a plane where all corners are equal and all sides are equal is called regular polygon.

12.Formula & Properties:

Sum of the inner angles of the triangle: The sum of the inner angles of the triangle is 180°

Nature of the outer corners of the triangle:

Property 1: One outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it.

Property 2: One outer angle of a triangle is greater than any of the inner angles that are not adjacent to it.

1. The sum of the inner angles of the polygon: the sum of the internal angles of the n-sided is equal to (n-2).

2. The sum of the outer angles of the polygon: the sum of the outer angles of the polygon is 360°

Method: There are oblique chords. When there is an oblique dust regret edge in the condition or solution, use the Zhengchang shed chord or cosine.

No oblique cuts. When there is no hypotenuse in the condition or solution, use tangent or cotangent.

Take the original and avoid the center. Try to use the original data and avoid intermediate approximations, otherwise it will increase the error of the final answer.

Rather multiply than remove. If you can use multiplication, you can use multiplication as quickly as possible, which can improve the accuracy of the calculation.

Knowledge points: 1. Two acute angles of a right triangle are surplus to each other.

2. The three high intersections of a right triangle are at one vertex.

3. Pythagorean theorem: the sum of the squares of two right-angled sides is equal to the square of the hypotenuse.

4. The sum of the inner angles of the right triangle is equal to 180 degrees, and the outer angles.

Method wide dismantling:

There are oblique chords. When there is an hypotenuse in the condition or solution, use sine or cosine.

No oblique cuts. When there is no hypotenuse in the condition or solution, use tangent or cotangent.

Take the original and avoid the center. Try to use the original data as much as possible to avoid the cautious approximation in the middle, otherwise it will increase the error of the final answer.

Rather multiply than remove. If you can use multiplication, you can use multiplication as much as possible to improve the accuracy of the calculation.

Knowledge points: 1. Two acute angles of a right triangle are surplus to each other.

2. The three high-crossed jujube points of the right-angled triangle are at one apex.

3. Pythagorean theorem: the sum of the squares of two right-angled sides is equal to the square of the hypotenuse.

4. The sum of the inner angles of the right triangle is equal to 180 degrees, and the outer angles.

1.The hot spots of trigonometric content propositions are the image and properties of trigonometric functions, solving triangles. (Every year, it must be envied only by hand).

2.The question structure of the test is basically one big question, one small question or three small questions, and the following three aspects are mainly examined:

1) the image and properties of trigonometric functions, which focuses on the images, monotonicity, maxima, periodicity and parity of trigonometric functions;

2) Trigonometric identity transformations, which focus on the induction formula, the basic relationship of the same angle trigonometric function, the sine and cosine formula of the sum and difference of two angles, and the formula of double angles;

3) Solving triangles, the focus is on using the sine theorem and cosine determination to understand the three mountains split angle or find the area and perimeter of the triangle.

Solution 3 makes this opening angle shape always a test point in mathematics, so what are the relevant and frank knowledge points for solving triangles? The following is a summary of the knowledge points I recommend to you to solve triangles, I hope it can help you.

Solution triangle definition:

In general, high school history, the three angles of a triangle, a, b, and c, and their opposite sides, a, b, and c, are called the elements of the triangle. The process of finding several elements of a known triangle to the other is called solving a triangle.

Main methods:

Sine theorem, cosine theorem.

Common ways to solve triangles:

Knowing one side and two corners to solve a triangle: Knowing one side and two corners (set to b, a, b), steps to solve a triangle:

2.Knowing the diagonal solution of the triangle on both sides and one of them: When you know the diagonal of the two sides of the triangle and one of them, when you find the other corners of the triangle, you must first determine whether there is a solution, for example, in the middle, known, there is no solution to the problem.

If there is a solution, is it one solution or two solutions. The number of solutions is discussed in the following table:

3.Knowing the two sides and their angles to solve the triangle: Knowing the two sides and their angles (set to a, b, c), the steps to solve the triangle:

4.Known trilateral solution triangle: Known trilateral a, b, c, solution triangle'Steps:

Use the cosine theorem to find an angle;

From the sine theorem and a +b+c= , find the other two angles.

5.Determination of the shape of the triangle:

Judging the shape of the triangle, we should think about the relationship between the corners of the triangle, mainly to see whether it is a regular triangle, an isosceles triangle, a right triangle, an obtuse triangle, an acute triangle, and pay special attention to the difference between "isosceles right triangle" and "isosceles triangle or right triangle".

The positive and cosine theorems are used to transform the known conditions into edge-edge relations, and the corresponding relationships of edges are obtained through factorization and formulation, so as to judge the shape of the triangle.

Using the positive and cosine theorem to transform the known conditions into the relationship between the trigonometric functions of the inner angles, through the identity deformation of the trigonometric functions, the relationship between the internal angles is obtained, so as to judge the shape of the triangle, at this time, we should pay attention to the application of a + b + c= This conclusion, in the equation deformation of the above two solutions, generally do not reduce the common factor on both sides, and should move the term to extract the common factor, so as not to miss the solution.

6.General ideas for solving oblique triangle problems:

1) Accurately understand the meaning of the question, distinguish between what is known and what is sought, and accurately understand the relevant names and terms in the application problem, such as slope, elevation angle, depression angle, angle of view, quadrant angle, azimuth angle, direction angle, etc.;

2) Draw a figure according to the theme;

3) Summarize the problem to be solved into one or several triangles, establish a mathematical model through the reasonable use of sine theorem, cosine theorem and other relevant knowledge, and then solve it correctly