Common tangent of two circles（两圆的公切线）

Common tangent of two circles（两圆的公切线） Common tangent of two circles(两圆的公切线) Junior high school mathematics teaching two common tangent circle The first class of common tangent of two circles (a) Teaching target: (1) understand the two circle tangent length and other related concepts, master the two circle grandfather tangent length method; (2) train students' ability of induction and summarization; (3) through the method of "two circles grandfather's tangent length", infiltrate the thought of "transformation" to students Teaching emphasis: Understand the two circle tangent length and other related concepts, two round grandfather tangent method Teaching difficulties: Two round grandfather's tangent and two round, grandfather's tangent length, the student understood is not thorough, is easy to confuse Teaching activity design (I) practical problems (introduced) A lot of machines on the drive belt and the driving wheel and the wheel from the positional relationship between the US and two in a straight line and tangent to the image. (this is a simple mathematical modeling, mathematical understanding and practice) (two) the concept of common tangent of two circles 1, concept: Teachers guide students to self-study. Definition two round Grandpa tangent, and the internal common tangent of common tangent length: And two circle tangent line, called the common tangent of two circles. (1): two Grandpa tangent circle on the common tangent of the same side, this is called Grandpa tangent tangent. (2) the internal common tangent of two circles in tangent: on both sides, so parents called the common tangent. (3) the common tangent length: two common tangent point distance called tangent length. 2. Understanding concepts: (1) long and long tangent common tangent of what is the difference between the contact? (2) long and tangent common tangent of what is the difference between? (1) a similar local tangent length and tangent length concept, that is, the line length is long. But the common tangent of two circles, and the line is two point as the endpoint; length of tangent to a circle, and one end of the line the point is, the other end is round a bit. (2) is a common tangent line, while the common tangent length is two point Q line length, the former cannot measure, which can measure. (three) the relation between the location of two circles and the number of common cut lines Organize students to observe, concept, summary, cultivate the students' learning ability. Write teaching materials P143 exercises second questions. (four) application, reflection and conclusion In 1 cases, O1 / O2, known: the radius were 2cm and 7cm, the center distance of O1O2=13cm, AB, O1, O2, is the Grandpa tangent, respectively is A, B. and AB.: common tangent length Analysis: the thought of tangent properties first, so the link O1A, O2B, AO1O2B. in general to have a right angle trapezoid to decompose it into a triangle and a rectangle, and then its properties. (students analysis, teachers' instruction, standardized procedures) Solution: link O1A, O2B, O1A group AB, O2B group of AB. O1 O1C group of O2B, C, quadrilateral pedal, rectangular O1ABC, Hence O1C O2 O1C= AB, C group, O1A=CB. Rt and O2CO1 in the delta. O1O2=13, O2C=, O2B-, O1A=5 AB= O1C= (CM) Reflection: (1) transform the thought and construct the triangle; (2) master the method of adding auxiliary lines at first 2*, as shown in figure O1, the O2, the known cut to P, linear AB as the common tangent of two circles, A, B as the cutoff point, if PA=8cm, PB=6cm, AB and tangent length. Analysis: because the line AB is a delta APB, Delta APB in PA and PB, known for long, just to prove that PAB is a right triangle, and then according to the Pythagorean theorem, the solution to the problem. The PAB card is a right triangle, just for Delta APB one angle is 90 degrees (or have had two horns and 90 degrees), which requires the communication angle, so the P for common tangent of two circles as shown in figure CD, because AB is the common tangent of two circles, so the angle CPB= angle ABP, angle CPA= / BAP. / BAP+ / CPA+ / CPB+ for angle ABP=180 degrees, so the 2 / CPA+2 / CPB=180 / CPA+ / CPB=90 degrees, so that the angle APB=90 degrees degrees, so APB is a right triangle, the better solution. The solution: P as a common tangent of two circles CD AB, O1 and dreams are tangent to its O2, A, B as the cut-off point L / CPA= / BAP / CPB= / ABP And dreams / BAP+ / CPA+ / CPB+ / ABP=180. * 2 / CPA+2 / CPB=180 degrees L / CPA+ / CPB=90 / APB=90 degrees degrees In Rt APB, AB2=AP2+BP2 Description: the two circle, often undercut for common tangent of two circles, the two round of communication angle. (five) consolidation exercises 1, when the two circles leave, the grandfather tangent, the distance between the center and the two radii must be made up (A) the right triangle (B), isosceles triangle (C), equilateral triangle (D), all of which are not correct This question examines the difference between the grandfather's tangent and the grandfather's tangent length. (D) 2, Grandpa tangent refers to (A) and line two round all the Zuqie (B) and the distance between the two points (C) in the two round of the common tangent tangent on both sides (D) in the two round with the common tangent beside the common tangent Direct use of the definition of the grandfather tangent. The answer is: (D) 3, teaching materials, P141 exercises (omitted) (six) summary (organized by students) The concept of knowledge: the long common tangent of two circles, Grandpa tangent, and the internal common tangent tangent; Ability: generalization, generalization, and the ability to seek grandfather's tangents; Thought: "transforming" thought (seven) homework: P151 exercises 10, 11. Second class common tangent of two circles (two) Teaching target: (1) master two circle tangent angle length and the method of calculating the tangent angle and connecting line or common tangent; (2) cultivate the transfer ability, further develop the students' ability of induction and summarization; (3) the two circle method to calculate the tangent length further infiltration to the students the idea of "transformation". Teaching emphasis: Method for the circle angle between two common tangent length and the angle between the tangent line and tangent or heart. Teaching difficulties: The two round and two round internal common tangent internal common tangent of students understanding through long, easily confused. Teaching activity design (1) review the basics (1) common tangent of two circles tangent, and common tangent concepts: internal and external common tangent length. (2) the relation between the position of the two circles and the number of the common cut lines (two) application and reflection In 1 cases, (2 cases materials) * O1 * O2 and known: the radius was 4 cm and 2 cm, the center distance of 10 cm, AB is a O1 / O2 and the internal common tangent, respectively is A, B. Ask: the common tangent length AB. Organizing the students' analysis and transferring the method of the tangent length of the grandfather can not only train the students' ability to solve problems, but also train the students' ability to learn Solution: link O1A, O2B, O1A group AB, O2B group of AB. O1 O1C O2B O2B group, to extend the line to C, Then O1C=, AB, O1A=BC. Rt and O2CO1 in the delta. O1O2=10, O2C=, O2B+, O1A=6 ?o1c (cm). ?ab = 8 (cm) 反思: 与外离两圆的内公切线有关的计算问, 常构造如此题的直角梯行及直角三角形, 在rt?o2co1中, 含有内公切线长、圆心距、两半径和重要数量.注意用解直角三角形的知识和几何知识综合去解构造后的直角三角形. 例2 (教材例3) 要做一个图那样的矿型架, 将两个钢管托起, 已知钢管的外径分别为200毫米和80毫米, 求v形角α的度数. 解 (略) 反思: 实际问题经过抽象、化简转化成问题, 应用数学知识来解决, 这是解决实际问题的重要方法.它属于简单的数学建模. 组织学生进行, 教师引导. 归纳: (1) 用解直角三角形的有关知识可得: 当公切线长l、两圆的两半径和r + r、圆心距d、两圆公切线的夹角α四个量中已知两个量时, 就可以求出其他两个量. ,. (2) 上述问题可以通过相似三角形和解三角形的知识解决. 巩固训练 (三) 教材p142练习第1题, 教材p145练习第1题. 学生独立完成, 教师巡视, 发现问题及时纠正. 小结 (四) (1) 求两圆的内公切线, 转化 "为解直角三角形问题.公切线长、圆心距、两半径和三个量中已知任何两个量, 都可以求第三个量. (2) 如果两圆有两条外 (或内) 公切线, 并且它们相交, 那么交点一定在两圆的连心线上. (3) 求两圆两外 (或内) 公切线的夹角. 作业 (五) 教材p153中12、13、14. 第三课时 两圆的公切线 (三) 教学目标. (1) 理解两圆公切线在解决有关两圆相切的问题中的作用, 辅助线规律, 并会应用. (2) 通过两圆公切线在证明题中的应用, 培养学生的问题和解决问题的能力. 教学重点. 会在证明两圆相切问题时, 辅助线的引法规律, 并能应用于几何题证明中. 教学难点. 综合知识的灵活应用和综合能力培养. 教学活动 复习基础知识 (一) (1) 两圆的公切线概念. (2) 切线的性质, 弦切角等有关概念. 公切线在解题中的应用 (二) 例1、如图, ?o1和?o2外切于点a, bc是?o1和?o2的公切线, b, c为切点.若连结ab、ac会构成一个怎样的三角形呢? 观察、度量实验 (组织学生进行) 猜想 (学生猜想 ?bac = 90 degrees). 证明: 过点a作?o1和?o2的内切线交bc于点o. ?oa、ob是?o1的切线. ?oa = ob. 同理oa = oc. ? ma = ob = oc. ??bac = 90 degrees. 反思: (1) 公切线是解决问题的桥梁, 综合应用知识是解决问题的 关键; (2) 作两圆的公切线是常见的一种作辅助线的方法. 例2、己知: 如图, ?o1和?o2内切于p, 大圆的弦ab交小圆于c, d. 求证: ?apc,?bpd. 分析: 从条件来想, 两圆内切, 可能作出的辅助线是作连心线o1o2. Or Grandpa's tangent Proof: P for the common tangent of two circles MN. Dreams / MPC= / PDC / MPN= / B, L / MPC / MPN= / B / PDC, The proposed APC = / BPD. Reflection: (1) the common tangent of two circles after MN, the two circle circumference angle angle in the link. MN should pay attention to the "bridge" role. (2) this method permits equal angles is the use of known angle calculation. Development: (organizing student research and fostering students' awareness of the problem in depth) Known: figure, O1 and O2 / / P / O1 inscribed in a circle, the string AB and O2 / C point tangent to the circle. Whether there is: APC = BPC / PC / APB. /. Answer: APC = PC / BPC / split / APB. figure as the auxiliary line, method of proof steps to see the typical examples in 4. cases (three) practice Exercise 1, teaching material 145, practice second questions Exercise 2, as shown in Figure two, the circle inscribed in P, the big circle string AB, cut in small round C, the big circle string PD over C points Confirmation: PA, PB=PD, PC. Proof: P EF as the common tangent of two circles AB is a small dreams tangent, the cut-off point of C L / FPC= / BCP / FPB= / A. And dreams / 1= / BCP- / A / 2= / FPC- / FPB L / 1= / A= / D / 2 dreams, star delta PAC to PDB PA - PB=PD - PC R Explanation: it is very easy to solve this problem on the basis of the expansion of example 2 (three) summary Learn the common tangent of two circles, should grasp the following aspects 1, the axial symmetry of the round, two round outer (or inner) tangent intersection (if it exists) are in line. 2, the calculation of the length of the cut line is transformed into the solution of right triangle, so the main idea of solving the problem is to construct right angled triangle 3, commonly used auxiliary lines: (1) the two circle often considers the center line in all cases; (2) two round cut, often add internal common tangent; two circle tangent, often add Grandpa tangent. 4, we should have in-depth study of the problem of consciousness, and constantly reflect on, and constantly sum up (four) homework materials, P151 exercises, group 15, group B, 2. Inquiry activity Question: as shown in Figure 1, the two circles intersect the A and B, and the line CD and the two circles intersect at C, E, F, and D. respectively (1) with a protractor to measure out angle EAF and angle CBD size, according to the measured results, what is the relationship between you and EAF / CBD / guess the size, and prove your conclusions. (2) when the position of the line CD is shown in Figure 2, the conclusion of the above question can still be established, and the reasons are given (3) if the known "two circle intersection" is changed to "two circles outside cut to point A", and the rest of the condition remains unchanged (Figure 3), then what will the conclusion of item (1) become? And make a certificate Note: (1) (2) (3) a / EAF+ / CBD=180 degrees. Proved slightly (Figure auxiliary line). Note: the experimental data obtained from the measurement of operating problems, data analysis, conclude the conjecture, and prove that the conjecture is true. This is a method for discovering mathematics. (2), (3) the problem is on the (1) promotion and special item. The conclusion (3) if the problem CD and move to the two circle tangent to the point of C, D, and then the conclusion will be / CAD = 90 degrees.