12 multiple-choice questions, progressively harder.
What must a symmetry of a figure do?
Solution
Correct answer: B
A symmetry may exchange points, but every part of the image must cover the original figure exactly. A single matching corner leaves the rest untested.
An unmarked rectangle has four right angles, width 10, and height 4. How many lines of symmetry does it have?
Correct answer: D
The horizontal and vertical center folds each match the whole rectangle. A diagonal would pair a length-10 side with a length-4 side, which reflection cannot do. No other mirror pairs its corners and sides.
An unmarked square has four right angles and four equal sides. What is its rotational order about its center, including identity once?
Correct answer: A
The matching angles are 0∘0^\circ0∘, 90∘90^\circ90∘, 180∘180^\circ180∘, and 270∘270^\circ270∘. These are four distinct rotations; 360∘360^\circ360∘ repeats identity.
Which turn defines point symmetry about O?
Correct answer: C
Point symmetry means a half-turn about the named point matches the whole figure. A quarter-turn or reflection is a different test.
Why are 0∘0^\circ0∘ and 360∘360^\circ360∘ counted only once when finding rotational order?
Both leave every point at its starting position. Order counts distinct mappings in a full turn, so the same identity mapping is counted once.
A figure has rotational order 1 about C, counting identity. What does this tell you?
Order 1 means identity is the sole matching rotation about C. It does not describe a nontrivial rotational symmetry or determine the reflection symmetries.
A figure has finite rotational order 5 about O, including identity. Find its least positive matching angle.
Five equal least turns complete one full turn.
360∘÷5=72∘360^\circ\div5=72^\circ360∘÷5=72∘
Why does a quarter-turn about the center fail for an unmarked rectangle of width 10 and height 4?
The quarter-turn produces a rectangle of width 4 and height 10 in the same plane. The long sides occupy different positions, so the whole boundary does not match.
How many lines of symmetry does an unmarked circle of positive radius have?
Every line through the center reflects the fixed-radius condition onto itself. There are infinitely many such directions. An offset line moves the circle to a different center.
Which describes the least positive matching rotation of an unmarked circle about its center?
Every positive turn matches the circle. Halving any proposed least positive angle gives a smaller positive matching angle. Zero is not positive.
For the exact block T outline shown, which candidate reflection matches the whole figure?
The vertical fold pairs the left and right parts of the top bar and the two sides of the stem. The horizontal fold sends the wide top bar to a wide bottom bar that is absent.
A half-turn exchanges opposite corners of a rectangle and maps every side onto an original side. Which conclusion is correct?
Symmetry concerns the set of positions occupied by the entire figure. Points can exchange places, so a matching figure need not have all of its points fixed.
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