1. Multiple-choice question: This topic is entitled *** 12, with 5 points for each question and 60 points for each question. Only one of the four options given in each small question meets the requirements of the topic.

1. As shown in Figure 4, the diameter of circle O is AB=6, C is a point on the circumference, and BC=3 intersects with C..

The tangent L of the circle, if A is the vertical AD of L and the vertical foot is D, then ∠DAC = ().

A.B. C. D。

This analysis is obtained from the tangent angle theorem, therefore,

So choose B.

2. In,, are the center lines of the height and the hypotenuse respectively, but in this figure * * * has a triangle and similarity, then ().

A.0 B. 1 C.2 D.3

Analysis 2: sum, so choose C.

3. Two chords of a circle intersect, one chord is divided into two sections 12 and 18, and the other chord is divided into two sections, so the length of the other chord is ().

A.B. C. D。

Analytically, assuming that the lengths of two segments into which another chord is divided are obtained by the intersection chord theorem, the required chord length is. So I chose B.

4. As shown in the figure, in and, if and.

If the difference between the perimeters of is, the circumference of is ()

A. 25 BC

Analytically, the answer d can be obtained by using that the similarity ratio of similar triangles is equal to the circumference ratio.

5. The secant passes through two points and the secant passes through the center of the circle. If known, the radius is ()

4th century BC

Let the analytic radius be, which is obtained by secant theorem, so choose D.

6. As shown in the figure, it is the diameter of a semicircle, with points on the semicircle and points on the point.

And, if, then = ()

A.B. C. D。

Let the analytic radius be, then, then, therefore, choose a.

7. In, the top is a point, the area is, and the area of the trapezoid is, then the value of is ().

A.B. C. D。

Analysis, using the area ratio equal to the square of the similarity ratio, you can get the answer B.

8. Two circles with radii of 1 and 2 are circumscribed, and a circle with radius of 3 is tangent to the two circles, and a * * * can be counted as ().

A.2 B.3 C.4 D.5

Analysis of a * * * can do five, of which two are circumscribed, 1 inscribed, and two are circumscribed and inscribed, so choose D.

9. As shown in Figure A, the quadrilateral is an isosceles trapezoid, which is made up of four such.

The isosceles trapezoid can spell out a parallelogram as shown in figure b,

The degree of quadrilateral is ()

A.B. C. D。

Analysis, therefore, choose a.

10. As shown in the figure, in order to measure the hardness of metal materials, a high-strength steel ball is pressed at a certain pressure.

Press on the surface of this material, leaving a pit on the surface of the material, and now the pit is measured.

The diameter is 10 mm, and if the diameter of the steel ball used is 26 mm, the depth of the pit is ().

A. 1 mm b.2mm c.3mm d.4mm.

Analysis depends on the meaning of the question, therefore,

Therefore, choose a.

1 1. As shown in the figure, let there be two points inside, and =+

A.B. C. D。

The analysis is shown in the figure, assuming that, then.

According to the parallelogram law, so =,

You can come to the same conclusion. So, choose B.

12. As shown in the figure, cut a cylinder with a plane at a certain angle with the bottom to get an ellipse, then this ellipse is.

The eccentricity is ()

A.b.c.d is not the above conclusion.

In this analysis, a plane is used to cut the cylinder, and the length of the minor axis of the ellipse is the diameter of the cylinder cross-section circle. This concept was clarified. Considering that the plane of the ellipse forms an angle with the bottom, the eccentricity is selected. Therefore, a was chosen.

Fill-in-the-blank question: This big question has four small questions, each with 4 points, *** 16 points. Fill in the answers on the lines of the questions.

13. The cross-sectional shape produced by the truncated ball is _ _ _ _ _ _ _; The cross-sectional shape produced by cutting a cylindrical surface is _ _ _ _ _ _ _ _ _.

Analysis circle; Round or oval.

14. As shown in the figure, in △ABC, AB = AC, ∠ C = 720, ⊙ O passes through points A and B and.

Tangent with BC at point B, intersecting with AC at point D, connecting BD, if BC =,

So AC =

Parsing consists of known,

Solve.

15. As shown in the figure, it is the diameter, chord and intersection point.

If, then =

Parse the link, and then, again,

Therefore,

So ...

16. As shown in the figure, the value of r in the figure is.

be

An analytical solution can be obtained from the diagram.

Third, the solution: this big question is ***6 small questions, and the score is ***74. The solution should be written in words, proof process or calculus steps.

17. (The full score of this small question is 12)

As shown in the figure, the two tangents of are tangents, and are.

The last two points, if, try to find the degree.

According to the tangent angle theorem, the analytical relationship can be obtained.

.

18. (The full score of this small question is 12)

As shown in the figure, the extension line with diameter ⊙ intersects the extension line of the chord at the point.

For ⊙O, dot, dot, and,

Find the length of.

Analytic connection, the relationship between the central angle and the central angle corresponding to the same arc

Combined with the conditions in the problem,

Therefore, therefore,

From the secant theorem, so.

19. (The full score of this small question is 12)

It is known that in isosceles trapezoid ABCD, AD‖BC,

Ab = DC, the intersection D is the parallel line DE of AC, and the extension line of intersection BA is in

E-point verification: (1) △ ABC △ dcb (2) de? DC=AE？ BD。

Analytically proved: (1) ∵ quadrilateral ABCD is isosceles trapezoid, ∴ AC = DB.

∵AB=DC,BC=CB,∴△ABC≌△BCD

(2)∵△abc≌△bcd,∴∠acb=∠dbc,∠abc=∠dcb

∵AD‖BC,∴∠DAC=∠ACB,∠EAD=∠ABC

∵ed‖ac,∴∠eda=∠dac ∴∠eda=∠dbc,∠ead=∠dcb

∴△ADE∽△CBD ∴DE:BD=AE:CD，∴DE？ DC=AE？ BD。

20. (The full score of this short question is 12)

As shown in the figure, in △ABC, AB=AC, AD is the center line, P is the upper point of AD, and the extension line of CF‖AB and BP passes through AC and CF in E and F. Verification: PB =PE? PF。

Analytic connection, easy to prove

* Therefore,

Also for the male role of * * *,

Therefore, ∴∴

This proposition is proved again.

2 1. (The full score of this small question is 12)

As shown in the figure, the diameters are upper point, upper point and lower point,

The tangent line passing through this point intersects with the extension line of, and the point is

Midpoint, connecting, extending and intersecting at this point,

The extension line of the extension line intersects the point.

(1) Verification:;

(2) verification: the tangent of yes;

(3) If the radius of and is, the length of and.

Analysis (1) proves that: yes diameter, yes tangent,

. Again,

It is easy to prove that,

. .

Yes, the midpoint.

(2) Proof: link diameter.

In, (1) is the midpoint of the hypotenuse,

.. again.

Yes, tangent.

Yes, tangent.

(3) Solution: act on points through points.

By (1), I know.

From what is known, there is an isosceles triangle.

,,, That's.

A quadrilateral is a rectangle.

It is easy to prove.

The radius of is.

solve ...

In,,, and, it is obtained by Pythagorean theorem.

Solve (negative values are discarded).

[Or take the midpoint, link, and then. Therefore, it is easy to prove. By, easy to know,.

By, solution. According to Pythagorean theorem, we get.

, (negative). ]

22. (The full score of this short question is 14)

As shown in figure 1, a point divides a line segment into two parts. If so, then this point is called the golden section of the line segment. When studying a project, a research group linked the golden section with the "golden section line" and similarly gave the definition of the "golden section line": a straight line divides a graph with area into two parts, and the areas of these two parts are respectively, and then the straight line is called the golden section line of the graph.

(1) The research group guesses that if the point is the golden section on the side (as shown in Figure 2), then the straight line is the golden section. Do you think it is correct? Why?

(2) Please explain: Is the midline of the triangle also the golden section of the triangle?

(3) In further exploration, the research team found that if a straight line passes through a point, then the straight line passes through a point and connects it (as shown in Figure 3), then the straight line is also the golden section. Please explain the reason.

(4) As shown in Figure 4, the point is the golden section of the edge, and it is obvious that the straight line is the golden section. Please draw a golden section so that it does not pass through the golden section on each side.

Analyze the golden section of (1) straight line. The reasons are as follows: the height of the edge of the set is.

,,, So,

Because the point is the golden section of the edge, there is. Therefore.

Therefore, the straight line is the golden section.

(2) Because the midline of the triangle divides the triangle into two parts with equal area, at this time, that is, the midline of the triangle cannot be the golden section of the triangle.

(3) Because the height of both sides of ∴ and * * * is equal, there is

Let a straight line intersect a point. So ... So ...

, .

Because it's the same sentence.

So the straight line is also the golden section.

(4) The painting method is not unique, and now two painting methods are provided;

Drawing 1: If you answer the graph 1, take the midpoint of and draw a straight line through, at, then this straight line is the golden section.

Figure 2: If you answer Figure 2, take a point on the map and connect it, then point to the intersection and connect it, then this straight line is the golden section.