【People's Education Press】Junior High School Mathematics Grade 9, Volume 1: The Essence of Rotation – From Everyday Phenomena to Mathematical Abstraction

Imagine a snowflake landing on your palm, or a water turbine spinning rapidly in a torrent. Behind these phenomena lies a unified geometric principle. This lesson will guide you beyond intuitive observation, using mathematical language to define 'rotation,' and explore the fascinating property of shapes that remain unchanged during rotation.

I. Mathematical Definition of Rotational Symmetry

In geometry, rotation is not a chaotic motion but a precise transformation. According to the textbook definition:

Definition: If a figure, when rotated by angle $\alpha$ around a point $O$, coincides exactly with its original form, then it is said to have rotational symmetry of angle $\alpha$ about point $O$.

This definition marks our shift from a dynamic process (rotation in motion) to a static property (symmetry). For example, a water turbine blade rotating $120^\circ$ around its axis can coincide with its initial position—this is a classic case of $120^\circ$ rotational symmetry.

II. Observation and Induction: Key Elements of Rotation

By comparing architectural patterns (static) with mechanical blades (dynamic), we can identify three core elements of rotational transformation:

Rotation Center: The point that remains fixed in position throughout the rotation.
Rotation Direction: Clockwise or counterclockwise.
Rotation Angle: The angle formed between the line segment connecting a point to the center and its image after rotation.

III. Methodological Transfer: Combining Numbers and Shapes

When studying quadratic functions, we derived their properties by observing their graphs. In the study of rotational transformations, we similarly adopt thisnumber-shape integrationapproach: deducing geometric properties (numbers) by observing the trajectory of shapes (forms).

🎯 Core Principle: Properties of Rotation

1. Corresponding points are equidistant from the rotation center;
2. The angle between the line segments joining any pair of corresponding points to the rotation center equals the rotation angle;
3. The figure before and after rotation is congruent.

QUESTION 1

Which of the following everyday phenomena does NOT constitute a rotational transformation?

The movement of clock hands

The rotation of electric fan blades

The vertical motion of an elevator

The turning of a steering wheel

QUESTION 2

If a square is rotated about its center, what is the minimum angle required for it to coincide with itself?

45°

90°

180°

360°

QUESTION 3

Which of the following statements about the properties of rotation is incorrect?

The shape and size of the figure remain unchanged before and after rotation

The rotation center remains fixed in position during the transformation

The distance from corresponding points to the rotation center may not be equal

The angle between the line segments joining any pair of corresponding points to the rotation center equals the rotation angle

QUESTION 4

In the Cartesian coordinate system, what is the coordinate of point A(3, 4) after a $180^\circ$ rotation about the origin?

(-3, 4)

(3, -4)

(-3, -4)

(-4, -3)

QUESTION 5

What is the smallest angle by which an equilateral triangle can be rotated about its center to coincide with its original form?

60°

120°

180°

240°

Comprehensive Exploration: From Dynamic Motion to Geometric Optimization

[Interdisciplinary Application] As shown in the diagram, a ball rolls from the top A to point B of an inclined plane. (1) Write the function expression for the rolling distance $s$ (m) in terms of time $t$ (s) (Hint: $s = \bar{v} \times t$, $\bar{v} = \frac{v_0 + v_t}{2}$). (2) If the incline is 3 meters long, how long does it take for the ball to reach the bottom?

Solution:
(1) Assume the ball undergoes uniformly accelerated motion starting from rest, with acceleration $a$. Then $v_t = at$. Substituting into the hint: $\bar{v} = \frac{0 + at}{2} = \frac{1}{2}at$. Thus, $s = (\frac{1}{2}at) \cdot t = \frac{1}{2}at^2$. This forms a quadratic function model.
(2) The value of $a$ must be known. If $a = \frac{2}{3}$ (a common textbook value), then $3 = \frac{1}{2} \cdot \frac{2}{3} t^2 \implies t^2 = 9 \implies t = 3$ seconds.

[Geometric Extremum] In right triangle △ABC, ∠A = 30°, ∠C = 90°, and AB = 12. A rectangle CDEF is to be cut inside, with vertices lying on each side. Where should point E be placed to maximize the area of the rectangle?

Solution:
Let $CD = x$. Then in right triangle △ABC, $BC = 6$, $AC = 6\sqrt{3}$. Using similar triangles $\triangle BDE \sim \triangle BCA$, we get: $\frac{DE}{AC} = \frac{BD}{BC} \implies \frac{DE}{6\sqrt{3}} = \frac{6 - x}{6} \implies DE = \sqrt{3}(6 - x)$.
Area $S = CD \cdot DE = x \cdot \sqrt{3}(6 - x) = -\sqrt{3}x^2 + 6\sqrt{3}x$.
This is a downward-opening parabola. The maximum area occurs when $x = -\frac{6\sqrt{3}}{2(-\sqrt{3})} = 3$. At this point, $CD = 3$, meaning D is the midpoint of BC. Consequently,point E should be located at the midpoint of hypotenuse AB.

[Coordinate Symmetry] Which two points among the following are symmetric about the origin $O$?
A(-5, 0), B(0, 2), C(2, -1), D(2, 0), E(0, 5), F(-2, 1), G(-2, -1)

Solution:
Points symmetric about the origin satisfy $(x, y) \to (-x, -y)$. Checking each point:
The symmetric point of C(2, -1) should be (-2, 1), which is exactly point F.
Therefore, C and F are symmetric about the origin.

[Open-ended Question] Can you provide some real-life examples of planar figures undergoing rotation? What are the properties of such rotations?

Sample Answer:
Examples: Windmill rotation, clock hand movement, car steering wheel turning, merry-go-rounds, wrench tightening bolts, etc.
Properties:
1. The figure before and after rotation is congruent (identical in shape and size);
2. Corresponding points are equidistant from the rotation center;
3. The angle formed between any pair of corresponding points and the rotation center equals the rotation angle.