Math Group: Zhou Haijun
First, the teaching objectives
(1) Three Verticality Theorem and its inverse theorem all study the vertical relationship between straight lines. They are widely used in the calculation and proof of spatial graphics, so this part of knowledge must reach the level of understanding and application.
(2) Using computer to simulate sports, enhance intuition, stimulate students' learning motivation, cultivate students' spatial imagination and transform mathematical thinking methods; At the same time, cultivate students' observation ability, guessing ability and demonstration ability.
Second, the teaching focus and difficulties
Emphasis: the teaching of triple vertical theorem and inverse theorem, and the application of these two theorems.
Difficulties: the application of three vertical theorems and inverse theorems.
Third, teaching methods.
Combining teaching with practice, using computer-aided teaching
Fourth, the teaching process and instructions
1, review the old knowledge and reveal the topic.
When discussing or calculating the properties of three-dimensional graphics, we often encounter the problem of judging the verticality of two straight lines or finding the distance between points and straight lines. These problems can be solved by discussing the perpendicularity between a straight line and a plane or transforming them into in-plane problems, but it is cumbersome to do so. Can you find a way to directly judge the perpendicularity of two straight lines in space?
For example, in the cube ABCD-a1b1c1d1,
(1) Find the projection of the oblique line BD 1 in the plane AC;
(2) What is the positional relationship between the straight line BD 1 and the straight line AC?
(3) What is the angle between the straight line BD 1 and the straight line AC?
Solution: connect BD and AC at point O, connect parallel lines of BD 1 and DD 1 at point M, and connect MA and MC.
Then ∠MOA or the rest angle is the angle formed by the straight line AC and BD 1 It is not difficult to get that MA=MC and O is the midpoint of AC, so MO⊥AC, that is ∠ MOA = 90.
∴ The included angle between the straight line BD 1 and AC is 90.
By recalling the positional relationship among diagonal lines, projections, straight lines and straight lines, this paper reveals the internal relationship between the content to be learned in this lesson and the original knowledge, that is, reminding students that the purpose of this lesson is to sum up conclusions and discover theorems by using the mathematical knowledge they have learned, thus laying the foundation for the proof of theorems.
2. Analyze the theorem and get the inverse theorem.
① Analyze the key words in the theorem, and the computer flashes corresponding words and graphics, with the purpose of helping students better understand the theorem and deepen their impressions.
② After the theorem is proved, ask: If the known conditions "a⊥AO" and "a⊥PO" are interchanged, does the conclusion hold? The movement of displaying "a⊥AO" and "a⊥PO" sentences dynamically by computer stimulates students' interest in learning and enhances their ability to explore problems.
(3) Consistency between theorem and inverse theorem, and analysis of elements and uses in theorem. Through computer dynamic display, students' understanding of the two theorems is further deepened.
Three Verticality Theorem: If a straight line in a plane is perpendicular to a diagonal of this plane, it is also perpendicular to this diagonal.
It is known that PA and PO are perpendicular and oblique lines of plane α, AO is the projection of PO on plane α, and straight line A is in plane α, a⊥AO.
Evidence: a⊥PO.
Prove:
AO⊥a
a⊥PO
Inverse theorem of three perpendicular lines theorem: If a straight line in a plane is perpendicular to a diagonal line of this plane, it is also perpendicular to the projection of this diagonal line.
Summary 1: The geometric elements involved in the theorem are:
(1) a plane;
(2) Four straight lines: ① a straight line perpendicular to the plane; (2) the diagonal of the plane; (3) the projection of the diagonal on the plane; ④ A straight line on the plane.
(3) Three verticals: ① The vertical line is perpendicular to the plane; (2) The projection of straight line and diagonal line on the plane is vertical; ③ The straight line and diagonal line in the plane are vertical.
3. Application theorem
Example 1. In space quadrilateral ABCD, AB⊥CD, AH⊥ plane BCD, verification: BH⊥CD.
Proof: ∵AH⊥ plane BCD
∴AB's projection on BCD plane is BH.
∵AB⊥CD, CD is in the BCD plane.
From the inverse theorem of three perpendicular lines theorem, BH⊥CD.
The purpose of the example 1 is to ask students to master the usage of the theorem, and summarize the general steps of proving the verticality of a straight line by using the theorem: find two and find three proofs.
Example 2: In the known right triangle ABC, the angle A is a right angle, the PA⊥ plane ABC, BD⊥PC, and the vertical foot is D;
Evidence: AD⊥PC
Proof: ∫pa⊥ Plane ABC
∴PA⊥BA
Are you ∵BA⊥AC?
∴BA⊥ Plane packaging
∴AD is the projection of BD on the plane PAC.
∵BD⊥PC again.
∴AD⊥PC。
(Inverse Theorem of Three Vertical Lines Theorem)
Example 3: ABCD-a1b1c1d1in cube.
(1) How many * * s are there perpendicular to the diagonal A 1C on each side of the cube?
Note: It can be concluded that the outgoing line is vertical;
(2) Let O be the midpoint of BD, and E and F be the midpoints of A 1B 1 and B 1C 1 respectively. Proof: d1o ⊥ ef;
Solution: Method 1: It can be proved that the projection of D 1O on the plane A 1C 1 is B 1D 1.
And B 1D 1⊥ straight lines EF, EF⊥D 1O are known from the theorem of three vertical lines.
Method 2: it can be proved that EF⊥ plane B 1BDD 1.
(3) If point P is any point on BD, then A 1C and D 1P are not perpendicular.
Description: Through typical exercises, students can learn the Three Vertical Theorem from different figures and angles, and highlight the essential element of the object-the vertical line of the plane, so as to correctly understand the Three Vertical Theorem, master various variants of the Three Vertical Theorem and its application points, which is very beneficial to strengthen migration and further cultivate students' spatial imagination and logical thinking ability.
4. Exercise: Judge right or wrong, and explain the reasons:
(1) If the projection of a straight line and an oblique line on the plane is vertical, then the straight line and the oblique line are vertical;
(2) If the projection of a straight line and a diagonal line on a plane is not perpendicular to this plane.