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Geometry 9-12 / Week 02 / Monday
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Week 02 · Rigid Motions and Symmetry

Monday

A flat and its twin
// The flats slide, flip and turn
⏱ about 20 min

Monday: A Flat and Its Twin

Comet slides a painted flat along the chalk grid. Its twin, painted as a mirror image, waits on the other side.

"Slide it three squares right and five down," Wren says, reading the plan. "What do you notice about the corners?"

"Every corner moved the same way," Comet says. "Three right, five down. The flat did not turn or flip."

Nova projects the twin's outline over the grid, reversed. "Would you like a hint? The twin is this flat flipped over the y-axis."

"Same panel, mirror image," Wren says. "A point goes in, a point comes out. That is a function."

"A function on points," Comet says. "Our designer, write where every corner lands."

Comet and Wren slide a painted flat past its mirror-image twin on the chalk grid while Nova projects the outline.

A transformation is a function

A transformation takes every point of the plane as an input and gives one point as an output, the image.

Write the image of A as A′, read "A prime". A whole figure's image is the figure made of all its image points.

Three transformations keep every distance and every angle: translations, reflections and rotations. We call them rigid motions.

MotionShop wordRule on the gridWhat it keeps
translationslide(x + a, y + b)every distance and angle
reflection over the y-axisflip(-x, y)every distance and angle
rotation of 90° about the originturn(-y, x)every distance and angle
horizontal stretchstretch(2x, y)angles and distances change

A solved problem to study

Wren slides the flat with corners (1, 1), (5, 1), (5, 3), (3, 4), (1, 3) by 3 right and 5 down. Here is his work.

Rule: (x + 3, y - 5). A (1, 1) becomes A′ (4, -4). B (5, 1) becomes B′ (8, -4).

Check a length: AB is 4 units. A′B′ is 4 units. The slide kept it.

Why it works: every point moves the same distance in the same direction. Nothing bends and nothing stretches.

The flat on the grid in gold and, in blue, its image after the translation by (3, -5).
The flat in gold and its mirror-image twin in blue, reflected over the dashed y-axis.
WHERE DOES THE CORNER LAND?
  • Read the question.
  • Tap your answer.
Corner C of the flat is at (5, 3). It slides by (3, -5). Where does C′ land?
Corner D is at (3, 4). The flat flips over the y-axis to make its twin. Where does D′ land?
Corner B is at (5, 1). The flat turns 90° counterclockwise about the origin. Where does B′ land?
RIGID OR NOT?
  • Read the question.
  • Tap your answer.
Which transformation keeps every distance and every angle?
A flat slides 3 right and 5 down. Which word names this rigid motion?
Side BC of the flat is 2 units long. After the reflection over the y-axis, how long is B′C′?
StatementTrue or false?
A transformation gives one image point for every input point.?
A translation moves some corners farther than others.?
A reflection keeps every side the same length.?
A horizontal stretch is a rigid motion.?
WHY THIS EXERCISETelling rigid motions from stretches is the first test of whether two flats can match.
Corner E of the flat is at (1, 3). After the slide (x + 3, y - 5), what is the y-coordinate of E′? Type the number.
WHY THIS EXERCISEA rule turns a motion on the floor into arithmetic on paper.
Try it
On graph paper, draw the flat with corners at (1, 1), (5, 1), (5, 3), (3, 4) and (1, 3).
Draw its image after the slide (x + 3, y - 5), then its reflection over the y-axis. Label every image corner with a prime.
Draw the flat (1, 1), (5, 1), (5, 3), (3, 4), (1, 3) and its reflection over the y-axis. Mark the distance from each corner to the axis.

Strong start, designer. Tomorrow each motion gets a precise definition, and you learn why the rules work.