Mid-term examination paper of eighth grade mathematics

Eight grades mathematics first volume midterm examination questions.

First, multiple choice questions

1. The following operation is correct ()

A.B. C. D。

2. Among the following real numbers, the irrational number is ()

A.B. C. D。

3. The following judgment is wrong ()

Two triangles with two angles and one side are congruent.

Two triangles with two sides and an angle are congruent.

Two triangles with two sides are congruent with the median line on one side.

D one side of two equilateral triangles corresponds to the congruence of the same triangle.

4. As shown in the figure, point P is above the bisector AD of ∠BAC, and point E is PE⊥AC.

Given PE=3, the distance from point P to AB is ()

a3 b . 4 c . 5d . 6

5. As shown in the figure, if AB‖EF, CE=CA and E = are known, then

∠ the degree of ∠CAB is

A.B. C. D。

6. Given the ratio of the degrees of the two internal angles of an isosceles triangle, the degree of the vertex of this isosceles triangle is ().

ABC or d

Second, fill in the blanks

7. The picture on the right shows a sailboat made of puzzles, in which congruent triangles form pairs.

8. As shown in the figure, line segments AC and BD intersect at point O, and OA=OC. Please add a condition to make △OAB△ obsessive-compulsive disorder

This condition is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

9. As shown in the figure, AC and BD intersect at point O, ∠ A = ∠ D, please add another condition to make △ AOB △ Doc,

Your supplementary condition is.

10. As shown in the figure, the bisector vertically bisecting the line segment intersects with.

Point, link, then the degree is.

1 1. Yiling Yangtze River Bridge is a three-tower cable-stayed bridge. As shown in the figure, the longest steel cable is hung on the left and right sides of the middle tower, and the distance from the bottom of the tower column to the point is meters, so the length is meters.

12. As shown in the figure, the middle point is the upper point,,,

Du Ze.

13. It is known that if one of the acute angles is folded so that the vertex of the acute angle falls at the midpoint of the opposite side, and the folded seam intersects with another right angle and hypotenuse, the circumference will be.

14. As shown in the figure, triangular pieces of paper,

Fold the triangle along a straight line through the point so that the vertex falls on the point on the side.

If the crease is 0, the circumference of is 10 cm.

15. Write an irrational number greater than 2.

16. It's an equilateral triangle, on the sides, and it's a triangle.

Third, the calculation problem

17. Calculation

Fourth, draw a picture.

18. in recent years, the state has implemented the "village to village" project and rural medical and health reform. A county plans to build a designated medical station between Zhang Cun and Licun, and Zhang Cun and Licun are located on two intersecting highways (pictured). The medical station must meet the following conditions: ① the distance to the two roads is equal; ② The distance to Zhang and Li villages is equal. Please determine the location of this point by drawing.

Verb (abbreviation of verb) proves the problem

19. Known: As shown in the figure, it is the bisector of sum.

Verification:

20. Known: As shown in the figure, straight line and intersection point,,.

Verification:

2 1. As shown in the figure, in the isosceles Rt△ABC, ∠ ACB = 90, D is the midpoint of BC, DE⊥AB, and the vertical foot is E.

The extension line of BF‖AC intersection DE passes through point B and is at point F, which is connected with CF. 。

(1) verification: ad ⊥ cf;

(2) Connect AF, try to judge the shape of △ACF, and explain the reason.

22. As shown in the figure, in equilateral, the points are on the edge and intersect with the points.

(1) Verification:;

(2) Accuracy.

VII. Openness

23. As shown in the figure, the points on the side intersect at points respectively. There are four situations: ①, ②, ③, ④.

(1) Please choose two conditions as the topic and the other two as the conclusion, and write the correct proposition:

The condition of the proposition is sum, and the conclusion of the proposition is sum.

(2) prove the proposition you wrote.

Known:

Verification:

Prove:

Eight, guess, explore the problem

24. In a known quadrilateral,,,, rotates around a point, and its two sides intersect (or their extension lines intersect) respectively.

It is easy to prove when rotating around a point (as shown in figure 1).

When rotating around a point, does the above conclusion hold in both cases of Figure 2 and Figure 3? If yes, please give proof; If not, what is the quantitative relationship between line segments? Please write your guess without proof.

Reference answer

First, multiple choice questions

1.C 2。 B 3。 B 4。 A 5。 B 6。 C

Second, fill in the blanks

7.2 8.∠A=∠C，∠B=∠D，OD=OB AB‖CD

9.ao = do or ab = DC or bo = co 10. (No penalty will be deducted if you fill in 1 15) 1 1. 456.

12. 13. 10 or1kloc-0/4.915. For example (the answer is not unique) 16.

Third, the calculation problem

17. solution: the original formula = 1+ 5 (2 points for each correct answer of the last three numbers) 4 points.

= 1 1 5

= 5 6 points

Fourth, the proof questions

18. Draw a picture.

Draw a 3-point bisector.

Do perpendicular bisector at 3 o'clock.

19. Proof: Because it is the bisector of sum,

So ...

In and,

So ...

20. harmoniously,,, once again,

, 3 points

, 4 points

.6 points

2 1.( 1) proves that in the isosceles right triangle ABC,

∵∠ACB=90o,∴∠CBA=∠CAB=45。

∵DE⊥AB, ∴∠ Debt = 90, ∴∠ Debt = 45.

BF‖AC，∴∠ CBF = 90，

∴∠ BFD = 45 = ∠ BDE，∴ BF = DB ..................................................................................................................................

And ∵D is the midpoint of ∴cd=db BC, that is, BF = CD.

In Rt△CBF and Rt△ACD,

∴Rt△CBF≌Rt△ACD，

∴∠ BCF = ∠ CAD .......................................................................... 4 points.

And < = BCF+< = GCA = 90,

∴∠ CAD+∠ GCA = 90, that is, AD ⊥ CF; Six points.

(2) △ACF is an isosceles triangle.

Reason: According to (1), CF=AD, △DBF is an isosceles right triangle, and BE is the bisector of △ DBF.

∴BE divides DF vertically into two, that is, AF=AD.

∴CF=AF，

∴△ACF is an isosceles triangle ...................................... 10.

22.( 1) Prove that it is an equilateral triangle.