Plane transformations, coordinate reasoning and proof
| English | 中文 | Pinyin |
|---|---|---|
| rigid motion/ˈrɪdʒɪd ˈməʊʃn/ | 刚性运动 | gāng xìng yùn dòng |
| perpendicular bisector/ˌpɜːpənˈdɪkjʊlə baɪˈsektə/ | 垂直平分线 | chuí zhí píng fēn xiàn |
A decision before an answer
- A ninety-degree rotation changes (x,y) to (-y,x) around the origin. Changing only one sign is a reflection, not that rotation.
- Your goal: Transform coordinates while distinguishing rigid motions from dilation.
Read the relationship
- A translation adds a displacement to each point; reflection across the x-axis sends (x,y) to (x,-y). A 90° counterclockwise rotation around the origin sends (x,y) to (-y,x). These rigid motions preserve lengths and angles. Dilation by factor k about the origin scales coordinates and lengths by k when k is positive, preserving shape but generally changing size.
- Use distance, midpoint and slope as geometric evidence.
Rotate (3,2) 90° counterclockwise about the origin.
Apply (-y,x).
Use the defining rule
- Distance between two points follows the Pythagorean theorem: square the coordinate differences, add and take the root. The midpoint averages each coordinate. Slope compares vertical and horizontal change; perpendicular nonvertical slopes have product -1. Vertical and horizontal lines require separate treatment rather than division by zero.
- Use valid congruence and angle relationships rather than a diagram’s appearance.
Midpoint of (0,2) and (6,8):
Average corresponding coordinates.
Check the conditions
- A geometric argument starts from stated or marked conditions. Vertical angles are equal; parallel lines give corresponding and alternate-angle relationships. Triangle angles sum to 180°. SSS, SAS and ASA/AAS can establish congruence with proper correspondence, while two equal angles establish similarity but not equal size.
- Use valid congruence and angle relationships rather than a diagram’s appearance.
Point (2,-1) translated by (3,4) becomes (5,3); rotated 90° counterclockwise about the origin it becomes (1,2). Between (1,2) and (7,10), distance is √(6²+8²)=10 and midpoint (4,6). Two triangles with matching side lengths 3,4,5 are congruent by SSS; triangles with equal angles may instead differ by scale.
A line of slope 2 has a perpendicular nonvertical slope ____.
The slopes multiply to -1.
Apply the task format
- Construction reasoning uses a compass to transfer lengths and draw equal-radius arcs. An intersection of equal-distance arcs can locate a perpendicular bisector; it is not a guessed midpoint from a sketch. Coordinate evidence can support a proof, but one specially chosen picture does not establish a general statement for every triangle.
- Use valid congruence and angle relationships rather than a diagram’s appearance.
Follow the specified centre and order of transformations. Do not use a visually parallel or perpendicular pair as a stated condition.
Which answer fits this case?
Transform coordinates while distinguishing rigid motions from dilation
Two equal triangle angles prove congruence.
They establish similarity; lengths may differ.
Keep the distinctions
- rigid motion 刚性运动 — A transformation preserving distances and angles.
- perpendicular bisector 垂直平分线 — The perpendicular line through a segment’s midpoint.
- Transform coordinates while distinguishing rigid motions from dilation.
- Use distance, midpoint and slope as geometric evidence.
- Use valid congruence and angle relationships rather than a diagram’s appearance.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.