Coplanar, nonoverlapping hexagons - July 29-Aug. 2, 1996
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Regular hexagon ABCDEF has side AF in common with regular hexagon AFGHIJ and side BC in common with regular hexagon BCKLMN. All three hexagons are coplanar and nonoverlapping.If AB = 64, what is the value of JN?
We received a fair number of correct answers to this problem, many with good explanations, from Raffles Girls School and other points farther north - like England. Here's a representative sample in order of receipt:Zhang Weibin & Lim Rongxuan Grade: 7 School: Raffles Girls Secondary School The answer is 128. Since the figure is made up of 3 regular hexagons, each of the sides of the hexagon is 64. A hexagon is made up of 6 equilateral triangles, each of side 64. Figure ABMJ is half a hexagon and therefore 3 equilateral triangles. MJ is the base of 2 equilateral triangles and so the length of MJ is 64 multiplied by 2, which is 128. Therefore the answer is 128.
Zhang Huibin Grade: Secondary 1 School: RAFFLES GIRLS' SCHOOL (SECONDARY) Answer: The value of JN is 128. Solution: JN is a line that cuts a hexagon exactly in half. As all the angles of a regular hexagon are the same, therefore half a hexagon can be divided exactly into three equilateral triangles. Each side of an equilateral triangle is of the same length. JN is the length of two sides of the equilateral triangles, thus JN is 128 - (64+64).Felicia Tan May Lian Grade: Grade 7 School: Raffles Girls School (Secondary) Since AB = 64, each side of hexagon ABCDEF is equal to 64 because ABCDEF is a regular hexagon. As hexagon ABCDEF has sides of equal length with hexagons AFGHIJ and BCKLMN, all three hexagons have sides of length 64 units. JN is equivalent to the diagonal of any hexagon, which is equivalent to the sum of two sides of the hexagons. Since the length of one side is 64 units, the sum of the length of two sides will be 64+64 = 128. Therefore, the length of JN is 128 units.
Lauren Green Grade: Last year (is that grade 13?) School: NLCS (England) The exterior angle of a regular hexagon = 360/6 = 60 degrees. This is because the shape is 'closed': walking around its perimeter you will end up back where you started and facing the same way after 6 turns. Interior angle = 180 - 60 = 120 deg (angles on a straight line) I J N M --- --- / \ / \ H | |A---B| |L \ / \ / G---F C---K \ / E---D So angles JAF and NBC are 120 degrees. Since the figures are regular, line AB when extended bisects both the above angles, so angles JAH and NBL are 60 degrees. Therefore distance JN = AB + 2*(AJcos60) cos 60 = 0.5 all sides = 64 Therefore JN = 64 + 2*(0.5*64) = 128
Justin Lam Grade: 7th/8th School: Sequoia Middle School Since the hexagons ABCDEF and BCKLMN and AFGHIJ are all regular and AB=BC=AF=64, AJ=BN=64, and <BAF=<ABC=<ABN=<BAJ=120 degrees. (Each angle of a regular hexagon is (6-2)*180/6 = 120 degrees.) Therefore we have a trapezoid ABNJ with AB=BN=AJ=64 and <BAJ=<ABN=120 degrees and <AJN=<BNJ=60 degrees. Drop an altitude from A to JN that intersects at X and another altitude from B to JN that intersects at Y. Then we have a right triangle AJX with <AJN=60 degrees and <JAX=30 degrees, a rectangle ABYX with AB=XY=64, and another right triangle BNY with <BNY=60 degrees and <NBY=30 degrees. Using the fact that 30-60-90 right triangles have a side that is adjacent to the 60 degree angle that is half the length of the hypotenuse OR looking at the equilateral triangle that is formed by putting right triangles AJX and BNY (side AX and side BY overlapped), JX+YN = 64. Therefore, segment JN = XY+JX+YN = 64+32+32 = 128.
Jennifer Kaplan Grade: 6 School: Castilleja 1. I knew that each angle of a hexagon equals 120 (to find each angle you take the number of sides, subtract 2, multiply by 180, and divide by the number of sides). 2. If you were to split one of the hexagons into two trapezoids, two of the angles would be 120, and two of the angles would be 60, and three of the sides would be 64. 3. If you were to look at the trapezoid JABN, two of the angles would be 120 (<JAB and <ABN) because angle JAB plus angle JAF plus angle FAB must equal 360, since both angles JAF and FAB equal 120, then angle JAB must also equal 120 (same with angle ABN). The other two angles of the trapezoid would be 60 (<AJN and <JNB) because angle AJN + angle IJA = 180, since angle IJA equals 120, AJN must equal 60 (same with angle JNB). Also three of the sides (JA, AB, and BN) equal 64 because they are all sides of the hexagon. 4. If you were to compare trapezoid JABN with a trapezoid inside of one of the hexagons, you would find that they are congruent. 5. Then if you were to take a trapezoid inside of one of the hexagons and divide it into three equilateral triangles where each side equals 64, therefore the larger base of the trapezoid equals two of the bases of two of the equilateral triangles (64 + 64) or 128. 6. Since the two trapezoids are congruent, the length of JN equals 128!
Brian Gordon Dartmouth '92 Answer: JN is 128. A regular hexagon can "tile the plane." Therefore, J,A,B, and N are four points of another regular hexagon, one that is the same size as the other three. Therefore its diameter is the same as that of the other three hexagons. The diameter of a regular hexagon is twice the edge, and it's real easy to show by simply drawing diagonals to create six congruent equilateral triangles. Each of those has length 64, so the diameter is 128, since it's two sides' worth of length (note the fancy mathematical jargon there.) --bri
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