Đề tài Học hình học không gian qua 15 phút củng cố với anh ngữ

HỌC HÌNH HỌC KHÔNG GIAN QUA 15 PHÚT CỦNG CỐ VỚI ANH NGỮ Lời ngỏ : Hình học không gian là 1 phân môn học toán hay. Tuy vậy , để học sinh thích học ,chịu tìm hiểu hết chương trình là không dễ. Thực tế , tôi thấy, giờ học nào, tạo ra được tình huống mới, có vẻ tự nhiên, không áp lực, bọn trẻ tỏ ra rất thích học toán, và học toán giỏi, kể cả học sinh đã chọn thi khối C, D. Hưởng ứng nhu cầu bước đầu, dạy và học toán trên Anh ngữ do nhà trường đề ra, năm nay, tôi và các em khối 11 có phần củng cố bài hình học không gian bằng tiếng Anh, khoảng 5_ 15 phút sau hết, ở tiết bài tập, mỗi bài học, cũng là dịp tạo yêu cầu mới cho các em vui mà học . Có thể chia tài liệu nầy làm 3 phần : _ Phần 1 : Những bài toán về quan hệ song song. Bài toán đơn giản, số lượng ít, do các em cần phải làm quen khái niệm , thuật giải của phân môn mới , từ vựng mới. Những bài toán về quan hệ vuông góc. Bài toán phức tạp dần, phong phú hơn , được chuẩn bị nhiều hơn, học sinh viết tự tin hơn . _ Phần 2 : Trình bày 1 tình huống mới trong 1 tiết học đã thực hiện. _ Phần 3 : Một số đề bài tập tham khảo, thay đổi, cho tiết học sau. Có 3 bài toán do học sinh tập viết trong giờ hoạt động nhóm, tôi cũng xin đưa vào đây, ghi nhận năng lực và tính hiếu học của các em. Một số bài tập đề nghị nữa, chủ yếu lấy từ chương trình hiện hành, mục đích chính vẫn là học toán. Rất tiếc là thời gian có thể dành cho vấn đề còn eo hẹp. Tôi e rằng, chúng tôi còn nhiều lỗi trong biên dịch. Mấy lời trình bày. Kính mong quí Thầy chỉ dẫn thêm. Giáo viên Nguyễn Kim Thoa Unit 1 : PARALLEL RELATION Problem 1 : Given pyramid S.ABCD. a/. Find the intersections of and (SAB) and (SAC) b/. Find the intersection of pyramid S.ABCD and c/. Find the MN’s condition in order the intersect plane of pyramid S.ABCD and is a trapezium. Solution Is there any common point of and (SAB) ? _ Direction of ? b. Means of the intersection of pyramid and a plane ? a/. Find the intersections of and (SAB) and (SAC): On the analogy, call, we have b/. Find the intersection of pyramid S.ABCD and : In turn MP,PQ,QN,NM are intersections of and (SAB), (SBC), (SCD), (ABCD) Thus the intersection of pyramid S.ABCD and is a quadrangle MNPQ c/. Find the MN’s condition in order the intersection of pyramid S.ABCD and is a trapezium : MNPQ is a trapezium (1) (as SA//MP) (2) as Oppositely, if MN//BC then MN//PQ, as(agreed) Thus MN//BC is the MN’s condition in order the intersection of pyramid S.ABCD and is a trapezium. Problem 2 : On a plane (P), given parallelogram ABCD. Draw the parallel rays : Ax // By // Cz // Dt ( in the same direction of (P)). A plane ( Q ) intersects with Ax, By, Cz, Dt at A’, B’, C’, D’. a/. Prove (Ax,By)//(Cz,Dt) b/. Which shape is tetragon A’B’C’D’? c/. Prove AA’ + CC’ = BB’ + DD’ Solution: a/. (Ax,By)//(Cz,Dt): b/. The shape of tetragonal A’B’C’D’? (1) On the analogy, we also have (2) (1)&(2) A’B’C’D’ is a parallelogram c/. AA’ + CC’ = BB’ + DD’ : Put , then : O is the middle point of AC , O’ is the middle point of A’C’, so: AA’ + CC’ = 2OO’ On the analogy, we also have BB’ + DD’ = 2OO’ Therefore : AA’ + CC’ = BB’ + DD’ Problem 3 : Given a pyramid S.ABC. Call I a central point of SB. , O is the centre of ∆ABC. a/. Find the intersection of the pyramid and (OIJ). b/. , () // (OIJ). Call BC = x ( x > 0 ). Find x in order () intersect with pyramid S.ABC. c/. Find the intersection of the pyramid and () base on x Solution : a/. Find the section of the pyramid and plane(OIJ) : Call P0 the midlle point of BC. In (SBC): (Ménélaus) , ie: C is central point of BD ( thus : ) In (ABC): In ABP: ( by Ménélaus theorem) (thus ) In ABC: (by Ménélaus theorem) Thus: Thus the section of plane (OIJ) and the pyramid is IJKL b/. Find x in order ( ) intersect with pyramid S.ABCD : () // (OIJ) or () (OIJ) the section of () and each plane of the pyramid, if existing, must either parallel or coincide with the side of the plane section IJKL . (). The direction of the intersection of () and (ABC)is the direction of LK, not the direction both of AB and AC, so certainly, () intersect with both AB and AC. Call the intersection of () and AB, AC : N, R, the intersections of SB, SC, SA and () : P, Q, E, if existing, then : If then , and : , moreover, . Thus, if then () intersect with the pyramid S.ABCD As N and NB . N N Thus, if N move on AB then M move on BM0 , () intersect either BC or AC. Thus : () intersect the pyramid c/. Confute the intersection of the pyramid and () base on x 0< x < 1 M BC: the plane section is MNR. x = 1 : the section is CNR 1 < x < 4 M CM0 and : the section is NPQR. Espeacially, x = 2: : the section is KLIJ x = 4: : the section is SNR. ( ) 4 < x < 5 M M0M1: the section is QNE Cho hình hộp ABCD. A’B’C’D’. Tâm I: tâm bình hành ABCD.A’B’C’D’. Chứng minh: vectơ đồng phẳng Given a parallelepiped ABCD. A’B’C’D’. Call I centre - Unit 2 : PERPENDICULAR RELATION Program 1 : Given a regular tetrahedron ABCD. Call M central point of BC. Calculate the angle between 2 lines : a/. AM and CD b/. AB and CD Hình a. Hình b. Solution : a/. Calculate the angle between AM and CD : Draw MN//CD then is isosceles at A: b/. Calculate the angle between AB and CD : Call N,P central points of BD, AC In Thus: (AB,CD) = 900 Problem 2 : Given a regular prism ABC. A’B’C’ whose value of side is a, and a plane : . Find the intersection of prism ABC.A’B’C’ and plane. Solution: -ABC.A’B’C’ is a regular prism is perpendicular to any line in (ABC) -Call I, I’ midpoints of BC, B’C’. -Thus: is a rectangle and is a square -AI: the height of equilateral triangle ABC with AB= a -B’C’, C’B are diagonals of a square BCC’B’. Thus, -Call J central point of CC” : - - - is a right-angled triangle, with a right angled at I. (Homework ) the answer : Find the intersection of prism ABC.A’B’C’ and plane, known . Problem 3 : Given a pyramid S.ABCD whose bottom ABCD is a diamond. a/. Prove b/. Prove is an right-angled triangle Solution: a/. Prove Call H a intersection point of three perpendiculars of b/. Prove is an right-angled triangle Problem 5 : Given a regular pyramid S.ABCD whose bottom of is a square. a/. Calculate the angle between SA and (ABCD) b/. Calculate the angle between SB and AC Solution: a/. Calculate the angle between SA and (ABCD) (as S.ABCD is a regular pyramid) is a right triangle with right angle O Thus b/. Calculate the angle between SB and AC Problem 4 : On a plane (P), given rhombus ABCD. From 4 points A,B,C,D, draw the rays Ax, By, Cz, Dt that all are perpendicular to ( P) and in the same direction of (P). Plane ( Q) // BD and Prove that A’B’C’D’ is a rhombus. Solution: ABCD is a parallelogram , AB//CD, AD//BC. (1) On the analogy, we also have (2) (1)&(2) A’B’C’D’ is a parallelogram On the analogy, we also have : Because , A’B’C’D’ is a rhombus . Problem 6 : Given a regular prism ABC.A’B’C’ whose bottom is a equilateral triangle. Value of every side is a. D a central point of CC’. a/. Calculate the angle between DB and A’B’ b/. Calculate the angle between (DAB) and (ABC) Solution: Call M the midpoint point of AB is a equilateral triangle with side a Problem 7 : Given a regular prism ABC.A’B’C’ whose bottom is a equilateral triangle. Value of every side is a. D a central point of CC’. a/ Fine the value of the distance between A’ and (ABD) b/. Fine the value of the distance between A’B’ and BD Solution: Problem 8 : The base of a pyramid S.ABCD is a rhombus ABCD of side a with center O. a/. Find the distance between A and (SBC) b/. Caculate the angle between SA and (SBC) c/. Caculate the angle between (SAD) and (SBC) Solution: In AB = AD = a ( as ABCD is rhombus) is an equilateral triangle O is a central point of BD (as In a rhombus ABCD, is right angle at O Thus a/. Find the distance between A and (SBC) O is a central point of AC From O, draw The intersection of (SBC) and (SOE) is SE. From draw with right angle O as a height OE is right angled at O with right angled at O has a hieght OH: Thus b/. Caculate the angle between SA and (SBC) Thus Call c/. Caculate the angle between (SAD) and (SBC) Problem 9 : Given a prism ABC.A’B’C’whose bottom is an equilateral triangle. coincide with the central point of BC. Two plan holdind AA’ are perpendicular together. a/. What is the shape of the right section? b/. Calculate the angle between the side elevation and bottom of prism ABC.A’B’C’ ( known common side is AC ) Solution a. What is the shape of the plane section? (do ABC đều ) In draw a height HK, K. Thus is the plane section is isosceles right-angled at K with BC = a KH = a/2 is riaght angled at H, the height HK: In (ABC), draw HI is the height of AHC , J is cental point of CA As is equilateral triangle so b. Calculate the angle between the side elevation and bottom of prism ABC.A’B’C’: (as ) tan Problem 10 : Given a pyramid S.ABCD whose bottom is trapezium which is right-angled at A and D. AD = 2a, AD = DC = a. SA = a. Call E central point of SA. . Call () plane holding EM and perpencular with (SAD) a/.Find () b/. What is the shape of and S.ABCD? Calculate its area base on a and x. c/. Find the set of when M move on AD Solution Comfirm ( ) and Thus As given ()(SAD) Thus () is the plane holding EM and parallel with AB b. What is the shape of and S.ABCD? Calculate its area base on a and x. As given (as AB In (SAB): Call F midpoint of SB then in SAB (EMF) hold EM and (SAD), so (EFM) is () ()MN (with ) By Thalès theorem: In (ABCD): call J midpoint of AD,then JB = a CJ//AD. Therefore : is isosceles right-angled at J, JC = a and BM = x As Call I intersection of AD and BC so AI = 2 AD = 2a isosceles right-angled at A so MN = MI = 2a –x The plane section is a right-angled trapezium EMNF at E & M c. Find the set of when M move on AD with intersection EM. In (SAD), draw , then and In (SAD): Therefore,: circle with diameter ED or the circle inscribe . Limit : On AB, M move from A to B so, H move from A to D on the chord with dianeter ED Therefore: If M move on AB then H move on circle with diameter ED. STUDENT’S WRITTEN PROBLEM Bài 1 : ( Bài của nhóm em Mai_ 11Văn ) Given a pyramid S. ABCD whose bottom ABCD is a parallelogram with centre O. In tun call I, J central points of SA, SD. a/. Prove b/. Call M, N central points of AB, OJ in turn. Prove MN // (SBC) Solution: a/. Prove : As given: (according to nature of a central line) (as ABCD is a parallelogram) (transitily) (Thalès theorem) The two planes in turn contain two lines that do not have the sam parallel corresponding direction, thus they are paralled b/. Prove MN // (SBC) : ( nature of a parallelogram ) Thus (transitively) (saw) We also have Meanwhile, Thus ( Bài của nhóm em Mai_ 11Văn ) Bài 2 : ( Nhóm em Uyên _11 Hóa ) The horizontal base of a pyramid is an equilateral triangle ABC of side 20m.The vertex O of the pyramid is at a hieght of 10 above the base and OA = OB = OC. Giving each answer to the nearest, find: (i) the inclination of OC to the horizontal (ii) the inclination of the plane OAB to the horizontal (iii) the angle OB The point U on OB is such that OU = 2/3 OB. Find the length of AU, giving three significant figures in the answer. Suggested solution: (i) Given OA = OB = OC. Since , the inclination of OC the horizontal (ii) The inclination of the plane(OAB) to the horizontal: Now (iii) Now Given OU = 2/3 OB Observe that And By consine rule, (Thân Thị Phương Uyên – 11 Hóa) Bài 3 ( Nhóm em Ngọc Hà, Hoàng Dịu-11 Văn ) Given a pyramid S.ABCD whose bottom ABCD is a rhombus with side a and center I. a/. Prove that b/. In triangle SAC, draw IK SA at K. Calculate the length of IK c/. Prove that Solution a/. Prove that b/. In triangle SAC, draw IK SA at K. Calculate the length of IK In the rhombus ABCD, By a theorem cosin, c/. Prove that has and so is an equiteral traingle Moreover We have: ( Ngọc Hà, Hoàng Dịu-11 Văn ) SOME SIMILAR PROBLEMS 5/. Given a pyramid S.ABCD whose bottom ABCD is a square. a/ Prove and b/. Prove c/.Calculate IJ base on a 6/. Given tretrahedron ABCD. is a equilateral triangle of side a. , the value od the distance between D and BC is a, H is the midpoint of AH a/. prove b/. prove 7/. is a equilateral triangle of side a. , I is the central point of BC, D is a symatric piont of A to I. a/. Prove b/. Find the distance between I and SA c/. Prove 8/. Given a pyramid S,ABCD whose bottom ABCD is a square with center is O. SA=SB=SC=SD=AB=a. M is SC’s midpoint. a/. Calculate SO b/. Prove c/. Calculate OM and the angle between (MBD) and (ABCD) 9/. Given a retangular parallelepiped ABCD.A’B’C’D’ with AB = a, BC = b, CC’ = c a/. Prove b/. Calculate AC’ 10/. Given a pyramid A.ABCD whose bottom is an equilateral triangle. SH is the height of S.ABCD . Prove and 11/. Given a pyramid S.ABCD whose bottom is trapezium which is right angled at A, B. AB = BC= a, AD = 2a, Call AM = x( 0 < x < a ) a/. Find the intersection of b/. Calculate the area of the intersection of . 12/. Given tetrahedron S.ABC whose bottom is right-angled at B, . a/. Prove are right-angled b/. Calculate the distance between A and (SBC) c/. Calculate the value of 13/. Given a rectangular parallelepiped ABCD.A’B’C’D’ whose bottom is a diamond whose value of each side is a. a/. Find the value of the hight of a rectangular parallelepiped ABCD.A’B’C’D’ b/. Calculate the value of the line segment which perpendicular to A’C and BB’ 14/. Given a tetrahedron OABC. . a/. Prove is acute angled b/. Prove H is a orthocentric of c/. Prove 15/. Given a pyramid S.ABCD whose bottom is a square with centre O. . I is the central point of BC a/. Prove b/. In turn call OH, OK the hieghts of. Prove 16/. The base of a pyramid S. ABCD is a paralellogram ABCD, . M is the midpoint of the edge SA , N is the midpoint of the edge BC, I is the midpoint of the edge CD . Prove S,J,O are in the straight line 17/. Given a pyramid S.ABCD whose bottom is a square ABCD of side a. a/. Call M the central point of CD. Calculate b/. Calculate the value of the distance between A and (SBC) c/. Calculate d/. Draw and calculate the value of the line segment which perpendicular to AC and SD 18/. Given . In turn call I, J central points of AB, CD. a/. Prove b/. Prove c/. Calculate AB, IJ base on a, x d/. Prove I is equididistant from two plan (ACD) and (BCD) e/. Find relation between x and a in order to make 19/. Given a tetrahedron SABC. is a right angled at B. In turn call AH, AK the heights of a/. Prove b/. Call . Prove 20/. Given a pyramid S. ABCD whose bottom ABCD is a parallelogram with centre O. In tun call I, J central points of SA, SD. a/. Prove b/. Call M, N central points of AB, OJ in turn. Prove MN // (SBC) 21/. Given a pyramid S. ABCD whose bottom is a quare of side a with centre O. I is the cental point of BC. a/. Prove b/. In turn call OH, OK heights of Prove . Calculate HK. c/. In turn call M, N, P, Q central points of SA, SB, SC, SD. Calculate MNPQ’s area base on a. 22/. Given a pyramid S.ABCD whose bottom ABCD is trapezium ( AB // DC, AB =DC ). a/. Find the intersection of (SAC) and (SBD) b/. . Find c/. . Find the set of I when I move on SD. d/. . Draw at H . Prove . Find the set of H when E move on DC. 23/. Given a regular prism ABC.A’B’C’ of side a. Call D central point of CC’. a/. Calculate the angle between DB and A’B’, and the angle between (DAB) and (ABC) b/. Prove a pyramid D. ABB’A’ is a regular pyramid BẢNG TRA TỪ VỰNG : Acute Angle Area Center Central point Circle Chord Common Diameter Diamond Direction Intersect Intersectton Intersectton (point ) Height Line Line of intersection Midpoint Parallel Parallelepiped Parallelogram Plane intersec Prism → Regular prism Pyramid → Regular pyramid Quadrangle Ray Rhombus Rectangle Relation Space Square Surface of one side Tetrahedron → Regular tetrahedron Theorem Triangle → Right-angled triangle → Isosceles triangle → Equilateral triangle Trapezium góc nhọn góc diện tích tâm trung điểm hình tròn dây cung chung chu vi hình thoi phương, hướng giao nhau ( tương ) giao giao điểm đường cao đường giao tuyến trung điểm song song hình hộp hình bình hành Thiết diện thẳng Hình lăng trụ → lăng trụ đều hình chóp → hình chóp đều Tứ giác tia hình thoi hình chữ nhật quan hệ không gian hình vuông mặt bên tứ diện → tứ diện đều định lý tam giác Tam giác vuông → Tam giác cân → Tam giác đều hình thang I/. THE INTEGRAL CALCULUS 1/. Trigonometric function: Given a trigonometrical function: y = sin x + cos x a/. Find the determination of this function b/. Draw the graph of this function c/. Find the maxima and minima of this function 2/. Simple trigonometric equations Solving trigonometric equations: 3/. Regular trigonometric equations: Solving trigonometric equations EXERCISES 1/. Given a trigonometrical function: y = cos x a/. Find the determination of this function b/. Draw the graph of this function c/. Find the maxima and minima of this function in 2/. Solving trigonometric equations: 3/. Solving trigonometric equations 4/. Given this equation: 1/. Prove is a solution of this equation 2/. Solve this equation with the way call 5/. The value of angle in a right triangle is solution of this equation: Prove that is an isoscele triangle. 6/. Given a equation: a/. Solve this equation with b/. Find the value of m to make this equation have solution 7/. Find solutions of a equation: 8/. Given a equation: a/. Solve this equation with b/. Find the value of m to make this equation have solution 9/. Find solutions of each of the following: II/. SETS AND PROBABILITY 1/. Counting rules 1/. Every card in the deck of 52 has an equal chance of being selected. Find the probability of selecting: a/. a red card b/. a black Ace c/. the Jack of diamonds 2/. What is the a priori probability of a given number showing on a roll of fair, six-sided die selecting a male from 100 2/. Rearrangment 3/. Arrangement 4/. Combinatory III/.INDUCTION, ARITHMETIC PROGRESSION 1/ Prove 2/. Find the first tem and the arithmetical ratio of each of the following: 3/. Find the first tem and the denominator of each of the following: EXERCISES 1/. Prove that 2/. Prove that 3/. Prove that 4/. Find the first tem and the arithmetical ratio of each of the following: 5/. Find the first tem and the denominator of each of the following: 6/. Calculate 7/. Given arithmetic progression a,b,c. Sum of them is 21. Turn in add 2,3,9 to a,b,c, we have a denominator progression. Find a,b,c. IV/. CONTINUITY AND LIMITS 1/. Find the limits: 2/. Muster the continuity of this function in EXERCISES 1/. In problem 1 to 5 prove each of the relationships with the aid of a series whose general term is the given function 3/ In problem 1 to 5 find the limits 4/. In problem 1 to 11 find the limits 5/. Muster the continuity of this function in 6/. Muster the continuity of this function at V/. DERIVATIVE Find the derivative : EXERCISES VI/.THE DIFFERENTIAL CALCULUS 1/. Find the derivatives of each of the following