Splitting Hexagons - September 25-29, 1995
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Take a regular hexagon and split it into three identical parts (I've included one below to give you a hand). What shape is each part?_________ / \ / \ / \ / \ \ / \ / \ / \_________/Now take another regular hexagon and split it into six identical parts. Can you do it at least two different ways? What shapes are your pieces? Be as precise as you can when describing the shapes.Harder: Split a hexagon into six identical kites (you might have to look up what a kite is).
Sometimes it's hard to draw pictures to illustrate things, so you might want to explain your answers, saying where the lines are drawn and what things are connected. You might do this even if you do draw pictures, so that everything is clear.
(Remember to use only spaces, no tabs in your drawings since tabs don't always look the same on different machines.)
Annie says: Solutions
Pretty good job this week! Many people drew some really good pictures, which is hard with just keyboard characters. Others tried to describe in words where they were drawing line segments.
Three things become especially clear with both these methods. First, it's an excellent idea to begin by labelling the hexagon; then it's easier to say where the new segments go. Also, if you're providing directional clues, like "from the top right corner," it's a good idea to give the orientation of the hexagon, if only to say "as shown in the problem." And after you describe the segments you've drawn, explain what shapes you've formed, first because the problem asked you to, but also because then readers can be sure they understand what you did.
Special notice to Cindy Spering, who explained how to split the hexagon into six congruent parts in infinitely many ways.
Following are highlights, and the full list of names and solutions is also available. Be sure to check out the Sketchpad sketches attached to some names.
Ann Coulouris Grade: 8 School: Center for the Arts and Sciences Saginaw, MI The regular hexagon split into three identical parts forms three parallelograms: ______ /\ \ / \ \ / \_____\ \ / / \ / / \/____ / The regular hexagon split into six identical parts forms six triangles: ______ /\ /\ / \ / \ /____\/____\ \ /\ / \ / \ / \/____\/ Another way to split a regular hexagon into six identical parts is by forming six parallelograms: ______ /\ \ / \-----\ / / \_____\ \/ / / \ /-----/ \/_____/ The regular hexagon split into six identical kites forms six identical kites: ______ / | \ /- | -\ / - | - \ \ - | - / \- | -/ \___|__/
Sarah Asper-Smith, Molly Donaldson, and Vanessa DeRoux Grade 10? Juneau, Alaska
Kristy Giballa Grade 10 School: Mt. St. Joseph Academy, Flourtown, PA For this problem I named each of the corner points of the regular hexagon A-F going clockwise and the centerpoint X. I was able to divide it into three identical parts or rhombuses. I did this by connecting the segments XB, XD, and X,F. This constructed three equal parts. Next, I divided the hexagons into six equal parts in two ways. The first was by connecting each of the corner points to the centerpoint X. This constructs six equal isoceles triangles. The other way is by dividing it into three rhombuses as above. Then connect the points BD, DF, and FB with segments. This also divides the hexagon into six triangles. I was also able to solve the final problem. I divided it into six equal kites. I did this by connecting the midpoint X with segments to the points exactly between (midpoints) of each of the corner points.
From: RRAvery Durham Academy, Durham NC We decided to do the three parts of the problem using the same picture. We named the vertices of the hexagon A, B, C, D, E, and F in clockwise order. We also labelled the center of the hexagon G. 1. To split the hexagon into three identical parts, we drew three three line segments from A to G, from C to G and from E to G. Each shape would be a rhombus. 2. To split the hexagon into six identical parts, we considered two different pictures. a. Draw six segments from A to G, from B to G, from C to G, from D to G, from E to G and from F to G. Each shape would be an equilateral triangle. b. Draw six segments from A to C, from C to E, from E to A, from A to G, from C to G and from E to G. Each shape would be an isosceles triangle. 3. To split the hexagon into six identical kites, draw six segments from G to the midpoints of the sides of the hexagon. Each shape will be a kite.
Cindy Spering Grade 9 School: Mount St. Joseph Academy, Flourtown, PA 1) The result of splitting a regular hexagon into 3 parts is 3 congruent quadrilaterals. This is possible when every other vertex is connected to the center point by a line segment. 2) Six identical parts are formed by connecting every vertex to the center point. This results in 6 equilateral triangles. If the same is done, but is done by connecting the center point to a point equidistant from each vertex, and in the same direction (such as clockwise), 6 identical quadrilaterals are formed. 3) Six kites may be formed by connecting the center of the regular hexagon to the midpoint of each side of the hexagon.
From: "Judy M. Young" Christopher Eskildsen Grade: 9 School: College Park High School (I will assume you already have drawn the hexagon.) How to divide a hexagon into 3 equal parts: 1st way: Draw lines from the center of the hexagon to every other point. 2nd way: Draw lines from the center of the hexagon to the midpoint of every other side. How to divide a hexagon into 6 equal parts: 1st way: Draw a line to every point. 2nd way: Draw lines from the center of the hexagon to every other point. Then draw three lines from midpoints of three sides to the opposite (parallel) side. 3rd way: Draw lines from the center of the hexagon to every other point. Then draw three lines connecting these points. How to divide a hexagon into 6 equal kites: 1st way: Draw lines from the center of the hexagon to the midpoint of every side.
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