A Math Forum Project

Geometry Forum - Problem of the Week Solutions - "The Dot and The Line," Sept. 26-30, 1994 Annie says: An amazing response this week! The answers submitted are really wonderful, and it seems that everyone had some fun (I know we sure enjoyed reading the answers). The best answers for the party trick were "country line dancing" and "twister". Awesome :-) There are some other gems in there as well, and I encourage everyone to read all the responses - they show a great range of knowledge and reasoning. The best part of this week's problem is that there weren't right and wrong answers. We were interested in the explanations, not the "yes" and "no". I will be reposting the problem to the newsgroup, and will ask the general readership to contribute their answers. We'll see what they can come up with! One more thing: If you haven't read The Dot and the Line, I heartily recommend it. FizzBomb 1. Is it possible for the line to be both a line and an angle at the same time? Why or why not? Yes, a line can be a 180 degree angle, if a vertex point is chosen. 2. Could the line turn itself into a ray? It would have to end somewhere. If it folded into itself it would become a ray, as far as I can understand. 3. Is it possible, if the line bends itself at several points simultaneously, for it to become more than one angle at a time? Yes. Well, you need to name points for the vertices and some points that lie on the angles. Other than that, it's just how you say -- it bends itself at several points simultaneously. 4. What other twists and turns could you add to this plot? Could the line lie on a curved surface and become a parabola? 5. What might become the line's favorite party trick? Country line dancing? I apologize for that. 6. (Write your own question!) Could the line wrap himself around a cylinder and become a helix? Julie Black 1. Yes, it is possible for the line to be both a line and an angle at the same time because if the line is straight, its measure is 180 degrees, which also means it is a straight angle. 2. The line could only turn itself into opposite rays. To do this, the line would have to add a dot to itself, creating opposite rays. However, the line could never be just a ray because a line goes on to infinity and a ray has a definite endpoint. 3. Yes, it is possible for the line to bend itself at several points and become more than one angle at a time. However, the angles would share one of their sides with another angle. 4. The line loved the beautiful dot so much that he decided that he wanted to have more angles than the triangle, so he turned himself into a hexagon. 5. The line's favorite party trick is showing the many sides of his personality by turning himself into an octagon. 6. Though the line often makes itself into an octagon, it never makes itself into a polygon that has more than 8 angles. What 3 kinds of polygons does the line never turn into? Jennifer Abrahamsen 1. The line could not be a line and an angle at the same time unless you consider a straight angle an angle. A figure is an angle if and only if it consists of two noncollinear rays with a common endpoint. The rays are the sides of the angle. The endpoint is the vertex of the angle. An angle separates a plane into three parts, the interior of the angle, the exterior of the angle, and the angle itself. The straight angle formed by opposite rays does not fit this definition. Therefore, by most standards a line can not be both a line and an angle at the same time. 2. By pivoting on a point, the line can turn itself into an angle. 3. No, it is not possible. This is because each side of an angle must be a ray or a segment that is extended indefinitely. This is not possible if the line is bent several times simultaneously. If this occurs, some sides will be segments instead of rays. 4. The dot and the line have children. The dot and the line have Siamese twin rays. They formed a straight angle. They needed to be bisected at birth. 5. The line's favorite party trick is his pivot. He forms all the angles in the room. 6. What famous code is made from the cousins of the dot and line? Betsy McCormick 1. Yes, it is possible for the line to be both a line and an angle at the same time because the line can be a straight angle of 180 degrees. 2. A line can be a ray but a ray would be only part of a line because a ray is defined as a segment that is extended indefinitely in one direction. Rays have exactly one endpoint and can also be defined using betweenness of points. A line could be made up of two rays, called opposite rays, which are determined by any given point on a line that they share as a common endpoint. 3. Sure, if the line bends itself at several points simultaneously it can become more than one angle at the same time. It would no longer be the original line, though -- instead it would be many forming more than one angle. 4. Perhaps the plane that both the line and the dot lie on could be head over heels for the line. What would the line do then? The line can not really escape from the plane. 5. Some of the line's favorite tricks might be turning into a ray by bending at a point and rotating 180 degrees so that none of his friends would know where the line started or ended. The line could confuse the other guests by randomly introducing itself as a line, a ray, or a straight angle. 6. If our original line was perpendicular to line AB, could you find the value of x? \ / \ / \ / \ / \ / 3x - 6 \ 2x + 26 / \ / \ / \ / \ / \ Eric Wahl 1. Yes. In some geometries a line is equivalent to a straight angle of 180 degrees. 2. Yes, if it could fold back onto itself at a particular point. 3. Yes. 4. The line could shrink to a dot (equivalent to a sex change) and then the first dot leaves him/her. 5. Rearranging the angles to different measures. 6. The line could turn into a 3-D object by turning onto itself to make a circle and then stretching the circle into a sphere; similarly, the line could turn onto itself to form a square and then stretch itself into a cube. Louise and Julia 1. Our answer is no because an angle is made from two rays and the line is only one line. 2. Our answer is no because lines go two directions (ways) and rays go in one. 3. Our answer is no because a line can't be an angle. 4. Our answer is there are lots of twists because a line does not have to be straight. 5. It will turn into a circle and catch the dot. 6. Can a line turn into a dot, ever? Naomi, Hayes, Molly, Sarah, Ed, and Katies 1. If the line has an intersecting ray it can be both line and angle at the same time or if the line sufficiently bent and intersected itself it could be an angle where it intersects. 2. No, a line keeps on going forever in both directions. Yes, if the line needs glasses and hits something as it is traveling (in search of more interested dots) and bounces back, then a piece of it could become a ray. 3. Yes, if it bends and intersects. 4. It could incircle the dot. 5. The line's favorite trick is to become a squiggly circle and encircle the dot. It could make buildings etc. for it (which we would show you but don't know how to draw in e-mail yet). 6. How can you guarantee that the line will meet another line and forget about the dot? What "line" does a line use to pick-up a dot? Melissa Semego 1. Yes, it is possible for a line to be an angle when two lines intersect. 2. Yes, because when the point is on the line, it divides itself into opposite rays. 3. Yes, it is possible for a line to bend simultaneously at two points to form a third point, which is a triangle. If you have more than 2 points and the line bends in different directions, you would come up with various obtuse and acute angles. 4. Depending on the direction of the bends, other geometric designs could take form, such as squares, triangles, spheres, rectangles, polygons, etc. 5. The line would form the shape of a heart as its favorite party trick. 6. What could the dot do to show its appreciation for all the effort put forth by the line? The dot could intersect with the line to form a ray so they can go forever in all directions. Andrea Hill 1. Yes because a line is an angle with a 180 measurement 2. Not without major surgery 3. Yes 4. The line and the dot have a meaningful relationship for a long time, when the line runs into an old girlfriend and has an affair. The dot finds out and gets so mad she prints herself in a geometry book. The line wakes up the next morning and feels guilty. He misses the line and decides that he wants to spend the rest of his long never-ending life with the dot. He gets printed too and they become an exclamation point, happily ever after. 5. An excellent hula-hoop 6. Do think it's possible for the squiggle to become insane and stalk the dot for the rest of her round life? Adam Freedman 1. If a line reflects off an object then it will become both a line and an angle. 2. No, a line cannot become a ray for a line has no beginning or end and a ray has a definite beginning. Though a line can make a point visible, so in a sense it would be 2 rays. 3. Yes, as stated in question 1. 4. Hits a wall and becomes disfigured. 5. Writing its name. 6. If the lines have a point in the middle could it bend itself in half? Jennifer Ashkenazi 1. Yes, it can be a 180 degree angle. 2. Yes, if someone or something cut a hole or a slit in it to make an end. 3. Yes, if it falls and bends like this: 4. The line fell and became an angle. The dot fell in love with him because he was different. The jealous squiggle made the angle feel bad because the dot only loved him for his looks. The angle left the dot and went off with another angle. 5. It could make a curve. 6. What's the line's favorite party game? Twister. Julie Stein 1. Yes. A line is a 180 degree angle. 2. No. It needs the help of a friendly dot. 3. Yes. The line would be more attractive than ever. 4. The jealous squiggle comes back and tries to kill the line. The line and the dot escape, but the line realizes that the dot only liked him for his looks and was very superficial. The line leaves the dot and runs off and elopes with a nice ray. 5. Twister 6. Why do you think the line liked the dot? Matthew Levine 1. Yes, a line is a 180 degree angle. 2. It could turn itself into more than one ray but not one ray. 3. Yes if it bends more than once. 4. There was dot's sister point. 5. Twister 6. How could the squiggle have made the dot fall in love with him? Robert Akerlof 1. It is possible for a line to be both a line and an angle at the same time. For example, point P is a point on line L. Point P divides line L into two rays. The definition of an angle is two rays with the same origin. This fits that description. 2. The line cannot be a ray because a ray has one ending point and a line does not. 3. If the line bends at several points, it can become more than one angle. If a line bends at point P and point Q, then there are two rays originating from each of the points, which forms two angles. 4. After the line figures out how to turn itself into an angle, the squiggle become a sine wave and teaches the dot trigonometry. The dot is so impressed that she falls in love with the squiggle again and they get married and have two tangents. 5. The line's favorite party trick is to write the word "line" by bending itself. He likes to show people examples of theorems in the letters he creates. 6. If the squiggle is a sine wave, what is the slope of the line if it just touches the top of the wave? Sarah Downie 1) Yes, because a line is a 180 degree angle. 2) Only with a dot could he become a ray. Without her he is nothing but a line. 3) Probably, it would make their marriage more interesting. 4) One day the line and the squiggle meet again and it turned out that they were long-lost brothers. The squiggle was kicked out of the house when he was little because he was too curvy. 5) The line's favorite party trick is the limbo. 6) What else could the line change about himself to make the dot happy so she never leaves him? Seth Schreiberg 1. Yes, a 180 degree angle because a line is a 180 degree angle. 2. No, not a ray but it could become multiple rays by defining a point on the line -- it would make the line two rays. 3. Yes, because at different spots on the line there would be angles. 4. Dot's family decides that the squiggle is better for Dot because they think that Line will use Dot to make himself a ray. 5. Tying himself in a knot. 6. What might be the line's favorite game? Sonya Feinberg 1. Yes, it is possible because a line is a hundred and eighty degree angle. 2. No, a line is never a ray because it never ends. 3. Yes, obviously if it bends itself at several points than it can make several angles. 4. The squiggle was very upset about loosing the dot. He cut the angle in two art the vertex, so he was no longer interesting to the dot. She then decided she loved the squiggle again. 5. His favorite trick is spiking the punch. 6. Why did the line love the dot in the first place? David Vigil 1) I suppose maybe it could be. Technically, I don't think so because a line can't intersect itself to make an angle and it can't really bend itself in half. But if I were a line and everyone called me one - dimensional", I would do situps and get in shape. So I guess if someone caught me in the middle of a sit up, I'd be an angle. 2) Yes. Maybe if Dot got pregnant and had a little Endpoint. Little Endpoint would get a piggyback ride from Line, turning Line into two rays. 3) Yes, but I suppose Line would have to be a contortionist. I mean, an angle isn't too hard, but at Line's age it isn't a smart idea to be multiple angles. 4) Well, since Line is doing all this bending in to angles, he ends up in an accident and turns into line segments. This accident happened the night after his birthday party. He was so old they made him into a number line. Dot, in a midlife crisis, might run away and become a pixel in order to recapture youth. She might even marry and change her name to Dot Matrix. 5) Well, Line isn't invited to many parties anymore. People found out when they tried to talk to him that he just goes on for ever and ever. 6) How many outcomes, and what kind, would you have if a squiggle and the dot had children? List all shapes and characteristics. Sara Runyan, Kelly Tang, Faaiz Bashir and Kristina Almquit 1) I think it is possible for the line to be a line and an angle if you use your imagination. For a start, the line could be a zero degree angle, which is both mathematical and fits the story line. For another reason, I think that it is very possible for the line to be bent. For example, if you have a laser beam, otherwise a line, and it reflects off a satellite continuously, then it was a line that was bent to make an angle. 2) The line couldn't turn itself into a ray because he like to face everywhere. He wants to experience and make friends with all the numbers on the number line. 3) I think that you could bend a line in many ways, therefore making many angles. A line goes on for infinity, therefore an infinite amount of angles could be formed. Early Greeks created light among their buildings by reflecting light rays from the sun from mirror to mirror. 4) We could make the squiggle become jealous so he turns himself into a circle. 5) The line's favorite party trick could be to limbo with another line. They could each take turns being the "stick". 6) Why do you think the dot is the female and the line is the male? What would the line have to do to prove to you that he was more interesting than a squiggle? Pamchal Javandel 1) Yes, because all the line has to do is bend over it will be an angle. When it stands up straight it will be a line again. It can also be any degree of an angle because it just bends down more or less. 2) Yes, because it could put his arms over his head and then put them down to 45'. 3) Yes, it could bend in more than just at the hips, like at the neck, knee, etc. 4) You could turn up the feet and have a curl, and you can arch your back and neck. 5) To move along the floor like a swiggle worm. 6) Can the line become a perfect circle? No, because the top of the circle will be a little pointy because they didn't meet perfectly. Andrew Liu 1) Positive, it is possible to be both. Theoretically speaking, a line could be turned into an angle by simply bending it by physical means. Or this theory could work vice versa; bending an angle into a line. But speaking in real terms, a line is an 180' degree angle. Though, the dot was probably more interested in the angles less than 180' because she didn't look beyond what he merely was. That's why he had to change himself to get her attention. 2) Math-wise, no. Physically, no. But all the line would have to do has a name of 0o all the way up to 360o. If its name was 45o then it would be a line and 45o angle. That is one way. another way is if it happened to have a split personality. It could be a line and split into an angle. Or if a female line and a male angle had a child it would be half line and half angle. - It would not happen in real life, a line can't be an angle and an angle can't be a line. Yes, it could turn. 3) Yes, but it really isn't good to do that much bending at one time. Maybe he should go see a doctor , or maybe become a Chinese Acrobat. 4) I think that the dot should fall in love with a traversable network (because she can really get into him) with two odd nodes, but when she comes along, she adds another odd node, so they break up. Plus, she didn't want to change her name from dot to node anyway! 5) The limbo, because he always seems to win. ( considering that he is one-dimensional, he can go really low.) 6) What does the dot call the line? (like a pet name?) My answer = collinear. Danielle Rollick 1) Yes, if the line has a name of 0o all the way up to 360o. If its name was 45o then it would be a line and 45o angle. That is one way. Another way is if it happened to have a split personality. It could be a line and split into an angle. Or if a female line and a male angle had a child it would be half line and half angle. 2) Yes, if Mr. line decides to legally change his name to Mr. Ray. Or if a line decides to go in one direction only. Yes, if you cut the line off. -No, a line is a line and a ray is a ray. A line goes in two directions, a ray goes in only one. 3) Yes, there are a couple of different ways. If the line ends make angles then yes. 4) The line so loved the dot that he would write love letters to her and never give them to her. He didn't think he had a chance. In the story it says that he changed himself into an angle. The line/angle and the dot had three little kids!!! Or should I say exclamation marks. So that shows, even though he may look like an angle he's still truly a line. 5) Juggling some beautiful dots. 6) What happened to the squiggle? He got so mad that he started to unsquiggle. He was so confused and didn't know what had happened so he turned into the top of the question mark. "?" Students at Mount St. Joseph Academy 1. Is it possible for the line to be both a line and an angle at the same time? It is possible for the line to be both a line and an angle at the same time because a line is a straight angle that equals 180 degrees. (Caitlin Anderson, 9th grade) Yes, it is possible because if a line bends to form an angle, it still extends forever in two directions. (Jordan Walsh, 9th grade) The line could not be both a line and an angle. He would need to anchor himself at one point to bend, therefore turning himself into two rays. (Kim Biedermann, 10th grade) Yes, a line can be an angle because the definition of a line is that it contains 2 points and this is true in a 180 degree angle. (Jennifer Hays, 9th grade) 2. Could the line turn itself into a ray? If the line changed to an angle he could become two rays. Otherwise he could not because a line goes on forever and a ray has an endpoint. (Maria Szczesniak, 10th grade) No, it can't because a ray has an endpoint and a line goes on forever in both directions. How can you stop a line? (You can't!) But the answer is yes if you consider two opposite rays. (Donna Brennan, 9th grade) Yes, if the line ever got together with the dot the dot could serve as an endpoint causing a ray. (Sarah Egner, 9th grade) A line can only turn itself into a ray if it has a point which will serve as an endpoint. Since the dot and the line are now in love, the dot could become the endpoint. (Susan Garges, 9th grade) 3. Is it possible, if the line bends itself at several points simultaneously, for it to become more than one angle at a time? He could turn into as many angles as he wants. (Kim Biedermann, 10th grade) No it is not possible. a line cannot bend. To create an angle you'd need at least two lines. (Susan Garges, 9th grade) Yes, a line can have several angles if it bends itself simultaneously because there are many places and directions it can turn, forming many angles. (Kristin Burton, 9th grade) 4. What other twists and turns could you add to this plot? I wanted the squiggle and the line to become friends and ditch the dot. (Ashley Hall, 10th grade) The squiggle, after months of practice, can learn to do the tricks the line is able to do. The dot then can't decide who she likes better and decides to dump the two of them and falls in love with Mr. Plane!!! (Kristin Burton, 9th grade) The line could tie himself into a knot with all the twists and turns and not be able to get undone. (Kelley Schmidt, 9th grade) The line could turn on his side and disappear. (Kim Biedermann, 10th grade) The dot and the line could get married and make their home on a plane. Then the dot and line could have little segment children. (Susan Garges, 9th grade) The line could form a star or a wheel and keep rolling away.... (Jordan Walsh, 9th grade) 5. What might become the line's favorite party trick?" Twister! (Kelley Schmidt, 9th grade) Making a maze and seeing who can get through and being a "limbo bar." (Ashley Hall, 10th grade) Winning the limbo contest because he can bend back so far and be really close to the ground and he can slide under any limbo stick no matter how low it is. (Caitlin Anderson, 9th grade) Coiling around himself in a circle over and over again and pretending to be his wife, the dot. (Jordan Walsh, 9th grade) Turning himself into a "curly que" and bouncing up and down like an accordion. (Donna Brennan, 9th grade) Line dancing! (Moira Conway, 10th grade) 6. Some of our questions. Could the line cross the street if it were parallel to him? (Kelley Schmidt, 9th grade). If the dot and the line got married and had kids, what shape would they be? (Beth Waters, 10th grade) If the dot placed itself on the line, could the line be an angle? (Yes, because the line could make a vertex). (Bridget Coyle, 10th grade) What were the children's names? Answers: Lionel, Dotty and Angel. (Maria Szczesniak, 10th grade) What is the line's favorite dance. Line dancing. (Sarah Egner, 9th grade). If the dot is at some time not on the line, how many planes contain both the dot and the line? Answer: only one (Susan Garges, 9th grade). Is it possible for the line to be a perfectly shaped circle or square? Not unless he swallowed his feet! (Kim Biedermann, 10th grade). Is it possible for the line to form some of the complex figures shown in the book? Why or why not? What do you think the line and the dot's house looked like? (Jordan Walsh, 9th grade) Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page