Tuesday, July 28, 2009

David Hankin Questions, many of them good & big ones

• What is geometry?
• When we study geometric figures, what are we concerned with?
• When we study geometric figures, what are we not concerned with?
• What kinds of motion (transformation) keep a figure “the same”?
• What is the maximum number of these motions (isometries) are necessary to get from one figure to another if they have the same orientation? If they have different orientation?
• What minds of motion change a figure?
• What is an angle?
• Given a diagram, what do you think might be true? (rather than given a diagram and some true things, prove this other thing)
• What is symmetry?
• Why do we study almost exclusively symmetric figures in geometry?
• Is a line segment symmetric?
• Is an angle symmetric?
• Are triangles symmetric?
• What can we say about the points on a perpendicular bisector?
• What happens when we find the perpendicular bisectors of the sides of a triangle?
• When is the circum center ON (/in/out) a triangle?
• What happens when we find the angle bisectors of a triangle?
• How many lines of symmetry does a quadrilateral have?
• Is it possible for a figure to have more than four lines of symmetry?
• How many lines of symmetry does a circle have?
• How do I draw a symmetry line through a circle?
• What happens when we draw the diameter through a point on a tangent line?
• What happens when we draw perpendicular chords through a diameter?
• What happens when we draw progressively smaller and smaller perpendicular chords?
• Problem: Given a triangle and side lengths, find the lengths of the segments formed by the incircles.
• Problem: Given a quadrilateral and side lengths, find the lengths of the segments formed by the incircles.
• Must a quadrilateral have a circumcircle?
• Reflection Problem: Given two points A & B, and a line (below them), what is the shortest path from point A to the line and then to B?
• Reflection Problem: In pool/minigolf, where do you aim so that A bounces once (twice, 3 times, 4 times back to A!) before hitting B? When are these possible/not?
• What has translational symmetry?
• Translation Problem: Two towns on opposite sides of a 1 mile wide river. Where should we put the bridge (to minimize the distance)? What if there are two nonparallel rivers?
• What letters are preserved under 180 rotation (/half turn/point reflection)?
• What geometric figures are preserved under point reflection? (which have point symmetry?)
• What geometric property follows from the fact that the letter Z has point symmetry?
• Point Reflection Problem: two circles intersect. Draw through the point of intersection a line which creates congruent chords in each circle.

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