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Transcript
Regular Polygons [Key Idea 7] 1. Constructing polygons using Tools: The definition of a regular polygon is an equilateral and equiangular polygon. Equilateral is not a sufficient condition because rhombi are not regular. Equiangular is not sufficient because rectangles are not regular. Therefore, to be a regular polygon, the figure must have equal sides and equal angles. Identifying the characteristics of regular polygons is an important topic for students, but constructing them can be tedious. Therefore, Geometer's Sketchpad comes with a file of regular polygon tools for students to use. Once in Geometer’s Sketchpad, open the folder labeled "Samples," then the sub-folder labeled "Custom Tools," and then the file "polygons.gsp." You now have a set of tools that you can use to construct polygons that can be accessed through the Custom Tool button. Once you select a tool, you will see at the bottom of the screen instructions that tell you what to do to make the construction. Schoaff, 2004 Regular Polygons p. 5 For example, "3/Triangle (By Edge)" needs 2 points. The instructions will say "1. Match Point A" and then "2. Match Point B." If you click on "Show Script View," you can see the entire script for this construction. This may be useful for students who wish to learn how to make their own constructions" To use this tool, simply select it, move the cursor to the screen, and follow directions. 2. Constructing regular polygons using iterations: The easiest way to construct a regular n-sided polygon without using a custom tool is by inscribing the polygon in a circle. a. Construct a circle. Select the center of the circle, under the Transform menu Mark Center. b. Select the point on the circle and under the Transform menu choose Rotate. Enter the desired angle as 360/n, where n is the number of desired sides to the polygon. (Our example uses n = 9.) c. Connect the two points on the circle with a segment. Schoaff, 2004 Regular Polygons p. 6 d. Select the first point, and then under the Transform menu choose Iterate. When the Iterate dialog box comes up, click on the second point. Click on the Iterate button. e. The program will automatically show several iterations. To complete the polygon, press the + key. 3. Challenge: There are several methods for constructing equilateral triangles and squares. How many different ways can you construct an equilateral triangle or a square using the tools under the Construction and Transformation menus? What properties are you using for each construction? Schoaff, 2004 Regular Polygons p. 7 Triangle Definitions & Properties [Key Ideas 1, 4, 7] The definition of a triangle is a 3-sided planar figure. With only this definition, what properties can we determine? Exploration #1 Attributes of triangles from three points. Objective: To construct a triangle from 3 points and explore angle and side measures. a. Select the Point tool. Make three points by moving to three locations on the screen and clicking the mouse button. Select the arrow tool to continue. b. Construct a triangle by selecting all three points. Now under the Construct menu choose Segment. The points will be connected in the order chosen. Measure the angles of the triangle. [Select three points, then under the Measure menu choose Angle. Now unselect the three points by clicking anywhere on the screen or pressing the esc key.] Notice that the vertices are now labeled in the order in which they were selected, not necessarily the order in which they were created. Repeat selecting three points and measuring for the two remaining angles. d. Sum the angles of the triangle by choosing Calculate under the Measure menu. Choose the first measure, click on the +, choose the second measure, +, choose the third measure, then Okay.] A e. Change the size and shape of the triangle. Point at one of the vertices (the arrow turns sideways). Hold down the mouse button and drag the point on the screen. Release the button. m – ABC = 69.66° m – BCA = 46.08° m – CAB = 64.26° Notice that the measurements for the angles have changed. What stayed the same? C m – A B C + m – B C A + m – CAB = 180.00° B f. Measure the three sides of the triangle by selecting all three segments (not the vertices), and then under the Measure menu choose Length. What do you notice about the relationship between angle and side measures? g. What if you drag a vertex so that two angles are equal in measure? What happens to the sides? What if three sides were equal in length? What else could you learn from this construction? Write two possible theorems. Schoaff, 2004 Regular Polygons p. 8 Exploration #2 Attributes of a Skewed Triangle Objective: To construct a triangle with one vertex on a line parallel to the base and then to determine any special properties. a. Under the File menu choose New Sketch. This gives you a new screen to work with. b. Construct the base of the triangle using the Segment tool to draw a segment on the screen. c. Construct the third vertex on a line parallel to the base as follows: - use the Point tool to place a point anywhere on the screen, not on the segment; - choose the arrow tool and select the point and the line segment; - then under the Construct menu choose Parallel Line. The parallel line should now be selected (colored) because it was the last object created. Under the Construct menu choose Point On Segment. You should now have a colored point on the parallel line. This is a movable point — practice dragging it. d. Construct the rest of the triangle. [The movable point on the parallel line is already selected. Select the two endpoints of the segment . Now under the Construct menu choose Segment.] C m AB = 7.92 cm Distance C to AB = 3.63 cm Area ACB = 14.36 cm2 Perimeter ACB = 20.45 cm B A e. Measure the length of the base of the triangle by selecting the endpoints. This will automatically label the base AB . Measure the altitude of the triangle by selecting the movable point and segment AB, then choose Distance under the Measure menu. This will automatically label the movable point C. Measure the area of the triangle as follows: first select points A, B, and C, then under the Construct menu, choose Triangle Interior. This interior will now be selected (plaid), so under the Measure menu choose Area. Measure the perimeter of the triangle. Select the interior, then under the Measure menu choose Perimeter. (Make sure you deselect the area measure first.) Schoaff, 2004 Regular Polygons p. 9 f. Move vertex C along the parallel line by selecting point C, then holding down the mouse button and dragging it. Since it is a child of the parallel line, it is restricted to moving along this line. Dragging the vertex of a triangle like this is called skewing the triangle. What changes? What stays the same? Why? Write a conjecture. (A conjecture is something that you think might be true. A conjecture, once proven, becomes a theorem. Many students list definitions as conjectures, but we don’t ‘prove’ definitions.) Can you change the area without changing the perimeter? Describe how you might do this. Exploration #3 Constructing Equilateral Triangle An equilateral triangle is defined as a triangle with 3 equal sides. C m AB = 4.08 cm m AC = 4.08 cm B A a. To construct an equilateral triangle, start with a clean screen. With the segment tool, draw a segment. b. Select one endpoint and then select the other endpoint. Under the Construct menu, choose Circle by Center+Point. While the circle is selected, under the Construct menu, choose Point on Circle. Construct a segment from the center of the circle to the new point. Measure both segments AB and AC. What do you notice? Can you explain why? c. Select point C and delete it. Select point B and then select point A. Under the Construct menu, choose Circle by Center+Point. Notice that the two circles intersect in 2 places. Using the arrow, point to one intersection and click. This places a point at the intersection. Select points A, B, and the intersection point, then Construct Segments. Measure the three segments to confirm that we have constructed an equilateral triangle using the definition (3 equal sides). Schoaff, 2004 Regular Polygons p. 10 m AB = 4.08 cm D m DA = 4.08 cm m BD = 4.08 cm B A d. Select the two circles, then under the Display menu choose Hide Circles. Measure the angles of the triangle. Make a conjecture. If ∆ABD is equilateral then Now drag point A or B to determine if your conjecture seems to hold. When you drag, what changes? What stays the same? (Note: you cannot drag D to change the size of the triangle because D is a child of the circles, which are children of points A and B.) 3B. Converse Is the converse true? That is, if ∆ABD is equiangular, is it also equilateral? (Are having 60˚ angles both necessary and sufficient for an equilateral triangle?) a. To determine this, start with a new segment. - Select one endpoint and under the Transform menu choose Mark Center (it will flash). - Select the other endpoint and under the Transform menu choose Rotate and then enter the angle 60˚. - Then select the first endpoint and the new point, in that order. Under Construct choose Ray. m – BAB' = 60.00° - Measure the angle to verify that it is 60. Hide to point on the ray. b. Now similarly construct a 60˚ angle at B by selecting B and under the Transform menu B choose Mark Center. Select point A and under the Transform menu choose Rotate and then enter the angle –60˚ (because we are rotating clockwise). Where did A' land? A Schoaff, 2004 Regular Polygons p. 11 c. Construct ray BA'. Measure angle ABA'. Hide the point A'. d. You should now have a triangle. Construct a point at the intersection of the two rays. Measure the length of segment AB. Measure the distance from A to the intersection point. Measure the distance from B to the intersection point. What kind of triangle do you have? (Note: you may want to discuss the difference in notation here, mAB as opposed to AB.) If ∆ABC is equiangular then Exploration #4 Properties of Medians of Equilateral Triangles A median in a triangle is a segment that joins the midpoint of a side of a triangle to the opposite vertex. a. Construct the medians of one of the equilateral triangles previously constructed. For example, using equilateral ∆ABC, select all three sides, then under the Construct menu choose Midpoints. Now connect each midpoint with the opposite vertex. b. Measure the lengths of the medians. What do you notice? In an equilateral triangle the medians are _____________________________. _ C D Measure the two angles formed at each vertex by the median. (For example, –ACF and –BCF in my figure.) What do you notice? E G In an equilateral triangle, the medians are also ____________ ______________________ A F B Measure the two angles formed by the median at the midpoint of the opposite side. (For example, –BDA and –BDC in my figure.) What do you notice? In an equilateral triangle, the medians are also ________________________________ Conclusion: If ∆ABD is equilateral then the medians Schoaff, 2004 Regular Polygons p. 12 Exploration #5 Constructing an Isosceles Triangle An isosceles triangle is defined as a triangle with two congruent sides. a. Start with a clear screen and a segment AB. We will use part of the construction for an equilateral ∆. C m AB = 4.90 cm m CA = 4.90 cm m BC = 7.74 cm _b. Select point A and then select point B. Under the Construct menu, choose Circle by Center+Point. While the circle is selected, under the Construct menu, choose Point on Circle. Construct segment AC. Measure both segments AB and AC. c. Construct segment BC. Measure this segment. Hide the circle. A d. Measure all three angles. What do you notice? Make a conjecture about the angles of an isosceles triangle. B If ∆ABC is isosceles then _____________ e. Can you drag the vertices of the isosceles ∆ABC until all three sides are congruent? If you can, that means that equilateral triangles are a subset of isosceles triangles. That is, all equilateral triangles are special isosceles triangles. f. Notice that in Exploration #3, when you drag a vertex of an equilateral triangle, it is always an acute triangle because all angles are less than 90˚. Can you drag a vertex of the isosceles triangle to make it acute? Can you drag a vertex of the isosceles triangle to make it obtuse? Can you drag a vertex of the isosceles triangle to make it a right triangle? Make a conjecture. If ∆ABC is equilateral, then it can be _____________, it can never be If ∆ABC is isosceles, then it can be , it can never be 5B. Converse: Is the converse true? That is, if a triangle has two equal angles, is it always isosceles? (Is having two equal angles both necessary and sufficient condition for an isosceles triangle?) a. To determine this, start with a new segment DE. - Select point D and under the Transform menu choose Mark Center. - Select point E and under the Transform menu choose Rotate and then enter any angle measure less than 90˚. Schoaff, 2004 Regular Polygons p. 13 - Select points D and the resulting point in that order. Under Construct choose Ray. - Measure the angle. b. Now similarly construct the same angle at E by selecting E and under the Transform menu choose Mark Center. Select point D and under the Transform menu choose Rotate and then enter the negative of the angle measure you used before. Construct the ray from E through the constructed point. Measure the angle. F m – E'DE = 75.00° m – D'ED = 75.00° FD = 10.36 cm FE = 10.36 cm c. You should now have a triangle. Construct a point where the two rays intersect and label it F. Measure the distance DF and EF. What kind of triangle do you have? If ∆DEF has two equal angles then ___________________________________. m AB = 6.39 cm m CA = 6.39 cm C m BC = 9.01 cm m DB = 7.13 cm D E m FC = 7.13 cm Exploration #6 Properties of Medians of Isosceles Triangles a. Construct the medians of the isosceles triangle ABC by selecting all three sides, then under the Construct menu choose Midpoints. Now connect each midpoint with the opposite vertex. Label the intersection. m EA = 4.54 cm D E G A b. Measure the lengths of the medians. What do you notice? B F Measure the two angles formed at each vertex by the median. (For example, –CAE, –BAE, –ACF, –BCF, etc. in my figure.) What do you notice? Measure the two angles formed by the median at the midpoint of the opposite side. . (For example, –AEC, –AEB, –BDA, –BDC, etc. in my figure.) What do you notice? Schoaff, 2004 Regular Polygons p. 14 If ∆ABC is isosceles then the medians to the equal sides _________________________ If ∆ABC is isosceles then the median to the unequal side ________________________ c. Is there any relationship between the lengths of the segments of the medians? In other words, using my labels, what is the ratio between AG :GE? The ratio BG :GD? The ratio CG : GF? Does this appear to be true for other kinds of triangles? 6B. Converse: d. You should have noticed that if ∆ABC is isosceles, then the median from the unequal angle (vertex angle) is also an angle bisector and altitude. Is the converse true? In other words, are any two of these attributes (median, angle bisector, altitude) sufficient conditions for an isosceles triangle? That is, if you construct a point on the ^ bisector of the base (median + altitude), will the triangle always be isosceles? Similarly, if you had any angle with a point on its angle bisector and you constructed a line through that point perpendicular to the angle bisector (angle bisector + altitude), would the resulting triangle be isosceles? A D 40.6° 40.6° 90.0° 90.0° D 90.0° 90.0° B C A AC = 3.68 cm BC = 3.68 cm Obviously, similar explorations can be done with other classifications of triangles, such as scalene, right, acute, obtuse, and combinations of these. Schoaff, 2004 Regular Polygons p. 15 Coordinate Geometry [Key Idea 1, 3, 4, 5, 7] Exploration #1 Reflections using Coordinates Under Edit Preferences, set the unit distance to cm. and units, set the angle to degree and tenths, and set the computations to tenths. Open a new sketch, from the Graph menu choose Show Grid and Snap Points. Select the two points at the origin and at (1,0) and under Display choose Hide Points. a. Place points on the grid to create an irregular polygon. Select the points in order and Construct Segments. Select the points and under Measure choose Coordinates. b. Select the y-axis and under Transform mark as Mirror. c. Select the entire polygon. Reflect. Measure the new coordinates. d. Compare the coordinates of the original points and the reflected points. Write a rule for reflecting in the y-axis. A': (-3.0, 6.0) B': (-3.0, 2.0) C': (-4.0, 2.0) D': (-4.0, 3.0) E': (-7.0, 3.0) F': (-7.0, 4.0) G': (-5.0, 4.0) H': (-7.0, 6.0) -10 H' A' F' A H 4 G' E' 6 G D' C' D B' 2 -5 B F E C A: (3.0, 6.0) B: (3.0, 2.0) C: (4.0, 2.0) D: (4.0, 3.0) E: (7.0, 3.0) F: (7.0, 4.0) G: (5.0, 4.0) H: (7.0, 6.0) 5 10 e. Undo the reflection and its coordinates. Select the x-axis and under Transform mark as Mirror. f. Select the original polygon. Reflect. Measure the new coordinates. g. Compare the coordinates of the original points and the reflected points. Write a rule for reflecting in the x-axis. h. Undo the reflection and its coordinates. Plot a point at (0,0) and (1,1). Construct the line through these points, which is the line y = x. Select this line and under the Transform menu mark it as a Mirror. i. Select the original polygon. Reflect. Measure the new coordinates. j. Compare the coordinates of the original points and the reflected points. Write a rule for reflecting in the line y = x. Schoaff, 2004 Regular Polygons p. 16 k. Undo the reflection and its coordinates. Select the origin and the line y = x. Construct a perpendicular. This is the line y = –x. Select this line and under the Transform menu mark it as a Mirror. 6 A H 4 G D 2 -10 B E C -5 F A: (3.0, 6.0) B: (3.0, 2.0) C: (4.0, 2.0) D: (4.0, 3.0) E: (7.0, 3.0) F: (7.0, 4.0) G: (5.0, 4.0) H: (7.0, 6.0) 5 10 -2 A': (-6.0, -3.0) B': (-2.0, -3.0) C': (-2.0, -4.0) D': (-3.0, -4.0) E': (-3.0, -7.0) F': (-4.0, -7.0) G': (-4.0, -5.0) H': (-6.0, -7.0) A' B' D' C' -4 G' -6 H' F' E' -8 l. Select the original polygon. Reflect. Measure the new coordinates. m. Compare the coordinates of the original points and the reflected points. Write a rule for reflecting in the line y = –x. Exploration #2 Translations using Coordinates a. Use the same set-up as in Exploration #1, including step (a). b. Select the entire polygon. Under the Transform menu choose Translate. Choose By Rectangular Vector. Enter a horizontal of –3 and vertical distance of –5. Translate. c. Measure the new coordinates. Schoaff, 2004 Regular Polygons p. 17 d. Compare the coordinates of the original points and the translated points. Write a rule for translating by a rectangular vector. 6 A H 4 G F D 2 B A' E C H' G' F' D' B' E' C' -4 D: (4.0, 3.0) E: (7.0, 3.0) F: (7.0, 4.0) G: (5.0, 4.0) H: (7.0, 6.0) A': (0.0, 1.0) 5 -2 A: (3.0, 6.0) B: (3.0, 2.0) C: (4.0, 2.0) B': (0.0, -3.0) C': (1.0, -3.0) D': (1.0, -2.0) E': (4.0, -2.0) 10 F': (4.0, -1.0) G': (2.0, -1.0) H': (4.0, 1.0) Exploration #3 Rotations with Coordinates a. Choose the origin and mark it as a Center. Use the same set-up as in Exploration #1, including step (a). b. Select the entire polygon. Under the Transform menu choose Rotate. Enter an angle of 90˚. c. Measure the new coordinates. d. Compare the coordinates of the original points and the rotated points. Write a rule for rotating by a 90˚. H': (-6.0, 7.0) G': (-4.0, 5.0) F': (-4.0, 7.0) E': (-3.0, 7.0) D': (-3.0, 4.0) C': (-2.0, 4.0) B': (-2.0, 3.0) A': (-6.0, 3.0) H' F' E' 6 A H G' D' A' C' 4 G B' D 2 B C F E A: (3.0, 6.0) B: (3.0, 2.0) C: (4.0, 2.0) D: (4.0, 3.0) E: (7.0, 3.0) F: (7.0, 4.0) G: (5.0, 4.0) H: (7.0, 6.0) e. Undo the rotation and coordinates of the new shape. Select the original polygon. Under the Transform menu choose Rotate. Enter an angle of 180˚. f. Measure the new coordinates. g. Compare the coordinates of the original points and the rotated points. Write a rule for rotating by a 180˚. Schoaff, 2004 Regular Polygons p. 18 h. Again undo the rotated polygon and its coordinates. Select the original polygon and 6 A 4 G D 2 B H A: (3.0, 6.0) B: (3.0, 2.0) F C: (4.0, 2.0) D: (4.0, 3.0) E: (7.0, 3.0) F: (7.0, 4.0) G: (5.0, 4.0) E C -5 H: (7.0, 6.0) 5 10 A': (-3.0, -6.0) C' E' B': (-3.0, -2.0) -2 C': (-4.0, -2.0) D': (-4.0, -3.0) B' D' F' E': (-7.0, -3.0) -4 F': (-7.0, -4.0) G': (-5.0, -4.0) H': (-7.0, -6.0) G' H' -6 A' under the Transform menu choose Rotate. Enter an angle of 270˚. i. Measure the new coordinates. j. Compare the coordinates of the original points and the rotated points. Write a rule for rotating by a 270˚. 6 A H 4 G F D 2 B E C 5 A: (3.0, 6.0) B: (3.0, 2.0) C: (4.0, 2.0) D: (4.0, 3.0) E: (7.0, 3.0) F: (7.0, 4.0) G: (5.0, 4.0) H: (7.0, 6.0) 10 -2 B' -4 C' A' D' G' -6 E' F' H' A': (6.0, -3.0) B': (2.0, -3.0) C': (2.0, -4.0) D': (3.0, -4.0) E': (3.0, -7.0) F': (4.0, -7.0) G': (4.0, -5.0) H': (6.0, -7.0) -8 Schoaff, 2004 Regular Polygons p. 19