Angles, Polygons, Circles
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- Blaise Weaver
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1 Page 1 of 5 Part One Last week we learned about the angle properties of circles and used them to solve a simple puzzle. This week brings a new puzzle that will make us use our algebra a bit more. But first, let s do a bit of review on angles and polygons. 1. What is the sum of the internal angles of any triangle? 2. What is the sum of the internal angles of any quadrilateral? 3. What is the sum of the internal angles of any pentagon? 4. What is the formula for the sum of the internal angles of any polygon? 5. What is the sum of the external angles of any polygon? Why do you think this is so? 6. A polygon with three sides we call a triangle. One with four sides is a quadrilateral, with five sides is a pentagon, with six sides is a hexagon. How many different types of polygons are there? Find the names and the sum of the internal angles for a: a. seven-sided polygon b. eight-sided polygon c. nine-sided polygon d. ten-sided polygon e. twelve-sided polygon 7. What is a regular polygon? What is the name for a regular polygon with the following number of sides, and what is the measure of each internal angle of that polygon? a. three b. four c. infinite (all of which must be infinitely short!) d. zero 8. Why is it impossible to have a polygon (regular or otherwise!) with only two sides? 9. All triangles tessellate. (To tessellate is to cover the plane using the same identical shape over and over again, without leaving any gaps.) Certain quadrilaterals tessellate (including squares and parallelograms). Certain pentagons tessellate (but not regular pentagons!). What other polygons tessellate?
2 Page 2 of 5 Part Two Given that the one labeled angle has value x, find where possible the value of all the other angles in terms of x. What angles can t you determine? You can assume that C is the center of the circle and that all of the line segments external to the circle are tangent to it at points A, B, D and E. AD and BE are diameters.
3 Page 3 of 5 Part Three You will need a compass, a ruler and a protractor for this exercise. Use the compass to draw a circle and a radius to that circle. Label the center C and the endpoint of the radius A. Open your compass so that it is precisely the same width as AC and draw a new circle with center A. Label the two points where the circles intersect B and D and draw a line segment between them. Label the intersection of AC and BD as Point E. Open your compass so that it is precisely the same width as BE (or ED, which should be the same length!). Draw a new circle, centered at B. Label the intersection with the original circle as Point F. Draw a line segment between B and F. Measure BF carefully with a ruler. Place your ruler at F and draw another line segment of the same length as BF. Continue on around the circle until you come back to B. If you measure accurately, you should come directly back to B. 1. What regular polygon have you just constructed? 2. Use your protractor to measure any of the internal angles. What is the result? Does this fit with your answer to the appropriate question in Part One? (To find out, multiply the result by the number of sides in the polygon. Because of rounding, you will have a small amount of error!) Deepest appreciation for the solution goes to Robin Hu. You can find more of his constructions at
4 Page 4 of 5 ANSWERS TO QUESTIONS Part One (n 2) * 180, where n is the number of sides of the polygon You remember that 360 is a circle, and when we go around the outside of a polygon we are going in a circle, are we not? 6. infinitely many! a. heptagon, 900. b. octagon, c. nonagon, d. decagon, e. dodecagon, A regular polygon is a polygon all of whose sides and angles are of equal measure. a. equilateral triangle, 60. b. square, 90. c. circle, undefined d. point, undefined Do you agree with our answers to c. and d.? Why or why not? 8. In Euclidean geometry the kind of geometry most of us are familiar with two lines (here we mean an infinite line, not a line segment, which is finite in length) can have one of the following properties: a. They can intersect in zero points, in which case they are parallel. b. They can intersect in one point. c. They can intersect in infinitely many points, in which case they are congruent (that is, they are the same line). There is no possibility for two lines to intersect in exactly two points. Therefore there is also no possibility for two line segments to intersect in exactly two points, and therefore there can be no twosided polygons. 9. The only other polygons that tessellate are regular hexagons. For extra credit, see if you can draw a tessellation with regular hexagons!
5 Page 5 of 5 Part Two x x 4. 2(90 - x) 5. 2x 6. 2(90 - x) x 16. 2(90 - x) 17. 2x 18. 2x 19. 2(90 - x) x 22. x 23. 2(90 - x) 24. x 33. 2(90 - x) 34. 2x 35. 2(90 - x) 36. 2x 45. 2(90 - x) Angle 24 must equal x, but the position of Point G is not fixed. Therefore angles 7, 8, 14, 15, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, and 44 will all change depending on the position of G. Similarly angle 11 must equal x, but the position of Point F is not fixed. Therefore angles 9, 10, 12 and 13 are subject to change depending on the position of F. Part Three 1. a heptagon 2. hopefully around 51.5!
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