It is known that the surface area of regular tetrahedron ABCD is S, and the centers of its four faces are E, F, G and H respectively. Let the surface area of tetrahedron EFGH be T, which is equal to ().
Asian Development Bank.
2. [2004 National College Entrance Examination (Shandong, Shanxi, Henan, Hebei, Jiangxi, Anhui volumes) Maths in Science 16, Maths in Liberal Arts 16]
It is known that a and b are non-vertical straight lines in different planes, so the projection of a and b on the plane may be.
① Two parallel straight lines ② Two vertical straight lines
③ The same straight line ④ A straight line and its outer points.
In a conclusion, the number of correct conclusions is (the number of all correct conclusions written).
3. [2004 national college entrance examination (Sichuan, Yunnan, Jilin, Heilongjiang) liberal arts mathematics question 6]
The length of the side and the length of the bottom of a regular pyramid are both 1, so the angle formed by the side and the bottom is ().
75-60 AD
4. [2004 National College Entrance Examination (Sichuan, Yunnan, Jilin, Heilongjiang) Question 7 of Science Mathematics, Question 10 of Liberal Arts Mathematics]
Given that the radius of the ball O is 1, three points A, B and C are all on the spherical surface, and the spherical distance between every two points is 0, then
The distance from the center o to the plane ABC is ()
Asian Development Bank.
5. [2004 national college entrance examination (Sichuan, Yunnan, Jilin, Heilongjiang) science mathematics 16, liberal arts mathematics 16]
The following are four propositions about quadrangular prisms:
① If the two sides are perpendicular to the bottom surface, the quadrangle is a straight quadrangle.
② If the sections of two opposite sides are perpendicular to the bottom surface, the quadrangular prism is a straight quadrangular prism.
(3) If four sides are congruent, the quadrilateral is a regular quadrilateral.
(4) A quadrilateral is a regular quadrilateral if its four diagonal lines are equal.
Among them, the number of true propositions is (the number of all correct conclusions written).
6. [2004 National College Entrance Examination (Shaanxi, Guangxi, Hainan, Tibet, Inner Mongolia) Question 9 of Science Mathematics, Question 10 of Liberal Arts Mathematics]
If the side length of the bottom surface of a regular triangular pyramid is 2 and each side is a right triangle, the volume of this triangular pyramid is ().
Asian Development Bank.
7. [2004 national college entrance examination (Shaanxi, Guangxi, Hainan, Tibet, Inner Mongolia) science mathematics 13, liberal arts mathematics 14]
If the distance from the center of the sphere to the plane is 0, then the ratio of the area of the truncated circle to the surface area of the sphere is 0.
8. [The third question of liberal arts mathematics in the national college entrance examination in 2004 (Gansu, Guizhou, Qinghai, Ningxia and Xinjiang)]
A diagonal line on the edge of a regular triangular prism is 2, which makes an angle of 45 with the bottom, so the volume of this triangular prism is ().
Asian Development Bank.
9. [2004 National College Entrance Examination (Gansu, Guizhou, Qinghai, Ningxia, Xinjiang) Science Mathematics Question 7]
For straight lines m, n and plane, the correct proposition in the following propositions is ()
A. if n is a straight line on different planes, then
B. If, n are straight lines on different planes, then they intersect.
C. If, n***, then
D. If, n***, then
10. [2004 national college entrance examination liberal arts mathematics 1 1 topic (Gansu, Guizhou, Qinghai, Ningxia, Xinjiang)]
It is known that the surface area of a sphere is 20, and there are three points A, B and C on the sphere. If AB=AC=BC=2, the center of the ball is flat.
The distance of surface ABC is ()
A. 1B。 CD . 2
11.[2004 national college entrance examination science mathematics 10 (Gansu, Guizhou, Qinghai, Ningxia, Xinjiang)]
It is known that the surface area of a sphere is 20π, and there are three points A, B and C on the sphere. If AB=AC=2 and BC=, then the center of the ball
The distance to the plane ABC is ()
A. 1B。 CD . 2
12. (Beijing college entrance examination in 2004, third in science and engineering, third in literature and history)
Let m and n be two different straight lines and three different planes, and give the following four propositions:
(1) If,, then
② If,,, then
(3) if, then.
(4) If, then
The serial number of the correct proposition is
A.① and ②B. ② and ③C. ③ and ④D. ① and ④
13.(2004 Beijing College Entrance Examination, Science and Technology, Question 4, Literature and History, Question 6)
As shown in the figure, in a cube, P is a moving point on the side. If the distance from P to the straight line BC is equal to the distance of the straight line, the trajectory curve of the moving point P is
A. straight line B. circle C. hyperbola D. parabola
14.(2004 Beijing College Entrance Examination, Science and Technology 1 1, Literature and History 12)
The latitude length of the north latitude on the globe is, and the radius of the globe is _ _ _ _ _ _ _ _ cm.
The surface area _ _ _ _ _ _ _ _ _ cm2.
15. [In 2004, the national college entrance examination (Shandong, Shanxi, Henan, Hebei, Jiangxi, Anhui volumes) ranked 20th in science mathematics, and 2nd in liberal arts mathematics, with a full score of 12].
As shown in the figure, it is known that the quadrangular cone P-ABCD, PB⊥AD, the side pad is a regular triangle with a side length equal to 2, the bottom ABCD is a diamond, and the dihedral angle formed by the side pad and the bottom ABCD is 120.
(i) Find the distance from point P to ABCD plane;
(II) Find the dihedral angle formed by surface APB and surface CPB.
16. [In 2004, the national college entrance examination (Sichuan, Yunnan, Jilin, Heilongjiang) ranked 20th in science mathematics and 20th in liberal arts mathematics, with a full score of 12]
As shown in the figure, in the prism ABC-a1b1,∠ ACB = 90, AC= 1, CB =, side AA 1= 1.
(i) Verify the CD⊥ plane bdm；;
(ii) Find the dihedral angle between B 1BD and CBD.
17. [In 2004, the national college entrance examination (Shaanxi, Guangxi, Hainan, Tibet, Inner Mongolia) ranked 20th in science mathematics and 2nd in liberal arts mathematics, with a full score of 12].
In the triangular pyramid P-ABC, the side PAC is perpendicular to the bottom ABC, and PA=PB=PC=3.
(1) Verification: AB ⊥ BC; ;
(2) Let AB=BC= and find the included angle between AC and PBC.
(2, liberal arts) If AB=BC=, find the dihedral angle formed by side PBC and side PAC.
18. [In 2004, the national college entrance examination (Gansu, Guizhou, Qinghai, Ningxia, Xinjiang) ranked 20th in science mathematics and 2nd in liberal arts mathematics, with a full score of 12].
As shown in the figure, in the quadrangular cone P-ABCD, the bottom ABCD is rectangular, AB=8, AD=4, the side pads are equilateral triangles, and the dihedral angle with the bottom is 60.
(i) Find the volume of the pyramid P-ABCD;
(ii) Proof of PA⊥BD.
19.(2004 Beijing college entrance examination, literature and history, 16, the full score of this small question is 14)
As shown in the figure, in a regular triangular prism, AB = 2, and the intersection point of the shortest path along the prism edge from vertex B to vertex is denoted as m, and the solution is:
(i) The diagonal length of the side development diagram of triangular prism.
(II) the total length of the shortest route
(III) The size of the dihedral angle (acute angle) formed by the plane and the plane ABC.
20.(2004 Beijing College Entrance Examination, Science and Technology, 16, the full mark of this small question is 14)
As shown in the figure, in a regular triangular prism, ab = 3, M is the midpoint, P is a point on BC, and the length of the shortest route from P to M along the edge of the prism is, let the intersection of this shortest route and N be, and find:
(i) The diagonal of the side development diagram of the triangular prism is longer.
(ii) Length of PC and NC
(III) The size of dihedral angle (acute angle) formed by plane NMP and plane ABC (expressed by inverse trigonometric function)
Reference answer
1.A2.①②④3.C4.B5.②④6。 C7.8.A9.C
10.A 1 1。 A 12。 A 13。 D 14。
15. [2004 national college entrance examination (Shandong, Shanxi, Henan, Hebei, Jiangxi, Anhui volumes) No.20 in science mathematics, No.2 in liberal arts mathematics1topic]
This small topic mainly examines the basic knowledge such as pyramid, dihedral angle, the relationship between line and surface, and also examines the ability of spatial imagination, reasoning and operation. The perfect score is 12.
(1) Solution: As shown in the figure, make a PO⊥ plane ABCD, the vertical foot is point O, and connect OB, OA, OD, OB, AD and PE at point E.
∵AD⊥PB,∴AD⊥OB，
∵PA=PD,∴OA=OD，
So OB bisects AD, and point e is the midpoint of AD, so PE⊥AD.
So ∠PEB is the plane angle of the dihedral angle formed by the curved surface PAD and the curved surface ABCD.
∴∠PEB= 120，∠PEO=60
From the known situation, PE=
∴PO=PE sin60 =，
That is, the distance from point P to plane ABCD is.
(2) Solution 1: Establish a rectangular coordinate system as shown in the figure, where O is the coordinate origin and the X axis is parallel to DA.
Link joint-stock company.
Also know from this:
therefore
Plane angle equal to dihedral angle,
therefore
So the size of the dihedral angle is.
Solution 2: Take the midpoint G of PB and the midpoint F of PC, connect EG, AG and GF, and then AG⊥PB, FG//BC, FG=BC.
∵AD⊥PB,∴BC⊥PB,FG⊥PB，
∴∠AGF is the plane angle of dihedral angle.
∵AD⊥ Facing Bobo, ∴ Advertising ⊥ eg.
And ∵PE=BE, ∴EG⊥PB, and ∠ PEG = 60.
In Rt△PEG, eg = PE COS60 =.
In Rt△PEG, EG=AD= 1.
So tan ∠GAE==,
And ∠ AGF = π-∠ GAE.
So the dihedral angle is π -arc tangent.
16. [In 2004, the national college entrance examination (Sichuan, Yunnan, Jilin and Heilongjiang) ranked 20th in science mathematics and 20th in liberal arts mathematics]
This small question mainly examines the basic knowledge of line-plane relationship and right-angle prism, as well as the ability of spatial imagination and reasoning operation.
The perfect score is 12.
Solution 1: (1) As shown in the figure, connect CA 1, AC 1 and CM, then CA 1=
∫CB = ca 1 =, ∴△CBA 1 is an isosceles triangle,
It is also known that d is the midpoint of its base A 1B,
∴CD⊥A 1B.∵a 1c 1= 1,c 1b 1=,∴a 1b 1=
And BB 1= 1, A 1B=2. ∫△a 1CB is a right triangle, and d is the midpoint of a1b.
∴CD=A 1B= 1，CD=CC 1，DM=AC 1=，DM=C 1M。
∴△CDM≌△CC 1M, ∠ CDM = ∠ CC 1m = 90, that is, CD⊥DM.
Because A 1B and DM are two intersecting straight lines located in BDM, CD⊥ plane BDM.
(2) Let F and G be the midpoint of BC and BD respectively, and connect B 1G, FG and B 1F, then FG//CD, FG=CD.
∴FG=,FG⊥BD.
BD = b1d = a1b =1can be known from the diagonal intersection of the side rectangles bb1a..
So △BB 1D is a regular triangle with a side length of 1.
So B 1G⊥BD, b 1g = ∴ b 1gf is the plane angle of dihedral angle.
And b1F2 = b1B2+bf2 =1+(=,
∴
That is, the size of the dihedral angle is
Scheme 2: as shown in the figure, establish a coordinate system with C as the origin.
(ⅰ)B(，0，0)，B 1(， 1，0)，A 1(0， 1， 1)，
d(，M(， 1，0)，
Then CD⊥DM. ∴ CD ⊥ A1B.
Because A 1B and DM are two intersecting straight lines in the plane BDM, CD⊥ plane BDM.
(ii) Let the midpoint of BD be G and connect B 1G, then
g()、、、)，
So the dihedral angle equals
17. [2004 National College Entrance Examination (Shaanxi, Guangxi, Hainan, Tibet, Inner Mongolia) No.20 in science mathematics, No.2 in liberal arts mathematics1topic]
This little question mainly examines the perpendicular nature of two planes, the angle formed by a straight line and the plane, logical thinking and spatial imagination. The perfect score is 12.
(1) Proof: As shown in figure 1, take the AC midpoint d to connect PD and BD.
Because PA=PC, PD⊥AC, also known as PAC⊥ ABC,
Therefore, ABC and D are vertical feet on the surface of PD⊥.
Because PA=PB=PC, DA=DB=DC,
It is known that AC is the diameter of the circumscribed circle of △ABC, so AB⊥BC.
(2) Solution of science: As shown in Figure 2, let CF⊥PB be in F and connect AF and DF.
Because △ PBC △ PBA, AF⊥PB, AF = CF
Therefore, PB⊥ Aircraft AFC,
So the intersection line of AFC⊥ pedestrian is CF,
Therefore, the projection of the straight line AC in the plane PBC is the straight line CF,
∠ACF is the angle formed by AC and plane PBC.
In Rt△ABC, AB=BC=2, so BD=
In Rt△PDC, DC=
At Rt△PDB,
In Rt△FDC, so ∠ ACF = 30.
That is, the angle between the AC and PBC planes is 30.
(2) liberal arts solution: because AB=BC, D is the midpoint of AC, BD⊥AC.
PAC⊥ ABC，
So BD⊥ plane PAC, D is vertical feet.
Make BE⊥PC in e, connect Germany,
Because DE is the projection of BE in the plane PAC,
So DE⊥PC and ∠BED are the plane angles of dihedral angles.
In Rt△ABC, AB=BC=, so BD=.
In Rt△PDC, PC=3, DC=, PD=,
therefore
Therefore, in Rt△BDE,
Therefore, the dihedral angle formed by the side PBC and the side PAC is 60.
18. [2004 National College Entrance Examination (Gansu, Guizhou, Qinghai, Ningxia, Xinjiang) No.20 in science mathematics, No.2 in liberal arts mathematics1topic]
This little question mainly examines the pyramid's volume, dihedral angle, angles formed by straight lines on different planes, spatial imagination and analysis.
Question ability. The perfect score is 12.
Solution: (1) As shown in figure 1, take the midpoint e of AD, connect PE, and then connect PE⊥AD.
Make a PO⊥ plane in ABCD, with the vertical foot of O and OE.
According to the inverse theorem of three perpendicular lines theorem, OE⊥AD is obtained.
So ∠PEO is the plane angle of the dihedral angle formed by the side pad and the bottom surface,
According to the known conditions ∠ peo = 60 and PE=6,
So PO=3, the volume of the quadrangular cone p-ABCD.
VP—ABCD=
(2) Scheme 1: As shown in figure 1, establish a spatial rectangular coordinate system with O as the origin. Can be obtained by calculation.
P(0，0，3)，A(2，-3，0)，B(2，5，0)，D(-2，-3，0)
therefore
Because of this, PA⊥BD.
Scheme 2: As shown in Figure 2, connect AO and extend AO to BD at point F. Through calculation, EO=3, AE=2,
We also know that AD=4 and AB=8,
get
So Rt△AEO∽Rt△ is not good.
Get ∠EAO=∠ABD.
So ∠ EAO+∠ ADF = 90.
So AF⊥BD
Because the straight line AF is the figure of the straight line PA on the plane ABCD, PA⊥BD.
19.(2004 Beijing college entrance examination, literature and history, 16, the full score of this small question is 14)
This small topic mainly examines the basic knowledge such as the positional relationship between straight line and plane, prism, spatial imagination, logical thinking and operational ability. 14.
Solution: (1) The side view of the regular triangular prism is a rectangle with a length of 6 and a width of 2.
Its diagonal length is
(2) As shown in the figure, rotate the side surface around the edge to make it on the same plane with the side surface. Point B moves to the position of point D, and the connecting line intersects with M, where M is the shortest path from vertex B to vertex C 1 along the edge of the prism, and its length is
therefore
(III) Connect DB, then DB is the intersection of plane and plane ABC.
auxiliary power unit
and
From the theorem of three vertical lines
Is the plane angle (acute angle) of the dihedral angle formed by plane and plane ABC.
The side is square.
Therefore, the dihedral angle (acute angle) formed by plane and plane ABC is
20.(2004 Beijing Institute of Technology College Entrance Examination 16)
This small topic mainly examines the basic knowledge such as the positional relationship between straight line and plane, prism, spatial imagination, logical thinking and operational ability. 14.
Solution: (1) The side development diagram of a regular triangular prism is a rectangle with a length of 9 and a width of 4, and its diagonal length is
(2) As shown in Figure 1, rotate the side surface around the edge to make it on the same plane as the side surface. When the point P moves to the position of the point, the connecting line is the shortest path from the point P to the point M along the side of the prism.
So, let's assume that it is obtained by Pythagorean theorem.
seek
(3) As shown in Figure 2, the connection line is the intersection of plane NMP and plane ABC, which is made in H, and the connection line between plane ABC and ch is obtained by the three perpendicular theorem.
Is the plane angle (acute angle) of the dihedral angle formed by plane NMP and plane ABC.
Yes,
Yes,
Therefore, the dihedral angle (acute angle) formed by plane NMP and plane ABC is