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Geometry 9-12 / Week 09 / Thursday
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Week 09 · Coordinates: Lines, Segments and Figures

Thursday

Proof Day: the platform is a rectangle
// The taped stage grid
⏱ about 20 min

Proof Day: The Platform Is a Rectangle

"Yesterday the numbers said rectangle," Wren says, pinning the lab log to the corkboard. "Today we write the proof."

"Given: four marks with coordinates," Comet says. "To prove: they make a rectangle. What goes first?"

"Parallel sides, from the slopes. That makes it a parallelogram," Wren says. "Then one right angle."

"Why only one?" Comet asks. "We checked all four yesterday."

Nova projects the parallelogram with one corner glowing. "Would you like a hint? Week 4: consecutive angles of a parallelogram."

"They add to 180," Comet says. "So one right angle forces the rest. Nice."

"Then the diagonals as a check," Wren says. "Our designer, put the steps in order and finish it."

The claim

A grid with the platform marks (1, 1), (7, 3), (6, 6), (0, 4) joined into a quadrilateral.

Given: A (1, 1), B (7, 3), C (6, 6) and D (0, 4) on the stage grid.

To prove: ABCD is a rectangle.

The proof, read first

  1. Slope of AB = 1/3 and slope of DC = 1/3. Equal slopes, so AB is parallel to DC.
  2. Slope of BC = -3 and slope of AD = -3. Equal slopes, so BC is parallel to AD.
  3. Both pairs of opposite sides are parallel, so ABCD is a parallelogram.
  4. Slope of AB × slope of BC = 1/3 × -3 = -1, so AB is perpendicular to BC. Angle B is a right angle.
  5. Consecutive angles of a parallelogram add to 180° (week 4), so every angle of ABCD is a right angle.
  6. Therefore ABCD is a rectangle.

Check: a rectangle's diagonals are equal. AC = 7.07 and BD = 7.07, both the square root of 50. They match.

A coordinate proof is the same proof as week 4, with slopes standing in for parallel marks and distances for tick marks.

A point on a circle

A grid with the turntable circle, center (6, 4) and radius 3.

The turntable's center is at (6, 4) and its radius is 3 meters. Is the mark (9, 4) on its rim? Its distance from the center is 3. Yes.

Is (8, 6) on the rim? Its distance from the center is the square root of 8, about 2.83. Not 3, so no.

A point is on a circle exactly when its distance from the center equals the radius. That is a one-line coordinate proof.

THE RECTANGLE PROOF, IN ORDER
  • ?Slope of AB = slope of DC = 1/3, so AB is parallel to DC
  • ?Therefore ABCD is a rectangle
  • ?Both pairs of opposite sides are parallel, so ABCD is a parallelogram
  • ?Slope of BC = slope of AD = -3, so BC is parallel to AD
  • ?Consecutive angles of a parallelogram add to 180°, so every angle is a right angle
  • ?Slope of AB × slope of BC = -1, so angle B is a right angle
WHY THIS EXERCISEEach step uses only the step before it and a fact from an earlier week. That is what makes it a proof.
Equal slopes on two sides show that the sides are this. Type one word.
Slopes multiplying to -1 show that two sides are this. Type one word.
A quadrilateral with both pairs of opposite sides parallel is called this. Type one word.
USE THE TOOLS
  • Read the question.
  • Tap your answer.
The circle has center (6, 4) and radius 3. Does the point (9, 4) lie on it?
The circle has center (6, 4) and radius 3. Does the point (8, 6) lie on it?
A grid with a segment at (7, 3), (8, 7)In Wren's failed set, BC runs from (7, 3) to (8, 7). What is its slope?
A grid with a segment at (6, 4), (8, 6)How far is the mark (8, 6) from the turntable center (6, 4)?
StatementTrue or false?
A parallelogram with one right angle is a rectangle.?
Equal diagonals alone prove that any quadrilateral is a rectangle.?
A point is on a circle when its distance from the center equals the radius.?
The point (8, 6) lies on the turntable's rim.?
WHY THIS EXERCISEKnowing which facts are enough, and which are not, is the heart of writing a proof.
Draw the proof figure: the four marks, the two pairs of parallel sides marked, and the right angle at B.

Clear reasoning, designer. Tomorrow you measure perimeters and areas straight from coordinates, then review the week.

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