POLYGONS AND SYMMETRY
| EXPLORATION | TITLE |
| 6-1 | REFLECTION-SYMMETRIC FIGURES |
| 6-2 | ISOSCELES AND EQUILATERAL TRIANGLES |
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6-2P |
SHARK ATTACK |
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6-3 |
CONSTRUCTING PARALLELOGRAMS |
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6-4 |
CONSTRUCTING A KITE |
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6-5 |
CONSTRUCTING A TRAPEZOID |
| 6-M | MYSTERY QUADRILATERALS |
| 6-6 | ROTATION SYMMETRY |
| 6-7 | REGULAR POLYGONS |
EXPLORATION 6-1. REFLECTION-SYMMETRIC FIGURES
Objective: Determine properties of reflection-symmetric figures.
1. CREATE A NEW DOCUMENT with the variable refsym.
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2. Create a vertical line on the screen. Draw the left hand side of a face with polygons, circles, and arcs as shown to the right. Create an arc Press F3 and choose 2:Arc, move the pencil to a starting position, press ENTER, move the pencil a little more, press ENTER, and move the pencil one more time until the desired arc, and press ENTER. Move the endpoints of the arc if |
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you would like to adjust its length. Move the middle point of the arc if you would like to adjust its curvature.
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3. Reflect each of the objects over the line until your face is complete. 4. Adjust the size and shape of the ears, eyes, nose, and lips on the LEFT SIDE of the screen and observe the changes on the right side. For smoother movement, you may want to move the pointer on top of a point, press 2nd, HAND KEY to LOCK, and drag using the keypad. Press ESC to return to the pointer when you are finished dragging. |
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5. The vertical line is called a line of symmetry. The face is reflection-symmetric about this line.
State at least two properties of a figure that is reflection-symmetric (without using reflection or symmetric in your explanation).
| 6. Drag the symmetry line until it is oblique and the face looks distorted. Construct a segment with the middle of the eyes as endpoints. What geometric property will the symmetry line have on this segment? Verify this conjecture by making appropriate measurements and/or calculations. Describe your findings below. |  |
EXPLORATION 6-2. ISOSCELES AND EQUILATERAL TRIANGLES
Objective: Determine properties of isosceles and equilateral triangles.
1. CREATE A NEW DOCUMENT with the variable isostri.
2. Create scalene triangle ABC (with unequal side lengths), measure and label segments AB and AC as shown below to the left.
3. Construct the midpoint D of side BC. Construct the median of BC (the segment connecting A and D), the perpendicular bisector of BC, and the angle bisector of < CAB as shown in the diagram above to the right.
4. Drag the vertex A until the median, perpendicular bisector, and angle bisector coincide. What type of triangle results? Explain below how you determined this result.
5. Measure the base angles, < ACB and < ABC. Write an IF...THEN... statement below describing the condition in step four and your results in step five.
6. Create the perpendicular bisectors of the other sides of the triangle, segments AC and AB. Drag the vertices of the triangle until the triangle is reflection symmetric with respect to all of the perpendicular bisectors.
7. Measure the remaining side and angle in the triangle. What type of triangle results? Explain below how you determined this result.
EXPLORATION 6-2P. SHARK ATTACK.
Objective: Determine the ideal position within an equilateral triangle given a series of conditions.
1. CREATE A NEW DOCUMENT with the variable shark.
Guppy, Tadpole, and Goldfish beach surround Equilateral Sea, a perfect equilateral triangle. Tiger shark and his family swim together in the Equilateral Sea. When the family gets hungry, Tiger makes sure they stay in one place (so he doesn't lose them), and then finds food at each of the beaches. Tiger gets food at one beach at a time because he can only hold so much food in his mouth at a time. So he will get food at one beach, come back to share the food (small square inside the triangle), and then repeat the process at the other beaches.
2. Use rotations to construct Equilateral Sea, and place a point inside the triangle to represent the hungry sharks. Construct perpendicular lines from the sharks to each of the sides of the triangle and the intersection points on the beaches.
3. Where should Tiger place his family in the Equilateral Sea in order to swim the least distance for his three hunting trips? Explain your reasoning.
4. What if the sharks lived in Isosceles Sea? Would your results from step 3 change? Why or why not? Explain your analysis.
EXPLORATION 6-3. CONSTRUCTING PARALLELOGRAMS
Objective: Construct a parallelogram and rectangle. Modify the parallelogram into a rhombus and the rectangle into a square.
1. CREATE A NEW DOCUMENT with the variable pararect.
2. Create an oblique line and a parallel line approximately 3 cm away. Connect the points on each line with a segment as shown in the diagram below to the left.
3. Construct a point on the line to the left. Construct a line parallel to the segment passing through this point, and create a point of intersection as shown in the diagram above to the right.
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4. Label the vertices M, A, T, and H. Construct overlapping segments and hide the three lines until your parallelogram matches the one to the right. It's quickest to hold down the shift key, select all three lines, and then choose the hide command. Remember to hit ESC after you hide the lines. 5. Measure the slopes of the sides of the figure. Drag the vertices around until the slopes change. Is the figure still a parallelogram? Explain how you know. |
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6. Delete the slope measurements. Measure the length of each side of MATH. Drag the vertices until each of the sides have the same length. This special type of parallelogram is called a rhombus.
Once you drag the vertices again, the parallelogram is no longer a rhombus because the sides will not have the same length. Your goal in chapter seven of the text is to construct a rhombus once you familiarize yourself with its properties.
7. Clear the screen. Create an oblique line and then construct a perpendicular with a point of intersection C. Place point U on one line and point S on the other line as shown below to the left.
8. Draw perpendicular lines passing thru points S and U. Label the point of intersection B as shown in the diagram above to the right.
9. Create overlapping segments CU, UB, BS, and SC. Hide the four lines. Choose Hide/Show, and then select the four lines by pressing ENTER.
10. Measure the length of each side of the figure CUBS. Drag points U and S until the measurements of all of the sides are equal. The most specific name for this figure is a square because all of its angles and sides are equal. This quadrilateral can be named a parallelogram, a rectangle, and a rhombus. Make necessary measurements on the calculator to justify why these names are also applicable to the figure. Explain below.
11. Once you drag the vertices again, the rectangle is no longer a square because the sides will not have the same length. Clear the screen and use rotations to construct a square that will always remain a square even after you drag around the endpoints. Explain the steps to your construction below.
Recall that you must create a numerical edit (under F7) for the number of degrees you wish to rotate the object.
EXPLORATION 6-4. CONSTRUCTING A KITE
Objective: Construct a kite using three different methods.
1. CREATE A NEW DOCUMENT with the variable kite.
2. Construct a kite using two intersecting circles as shown in Figure I of your text on page 323. Hide the circles and leave only the kite. Make necessary measurements on the calculator to justify why this is a kite. Explain below.
3. Clear the screen. Construct a kite using a reflected triangle over a symmetry line as shown in
Figure II of your text on page 323. Make necessary measurements on the calculator to justify why this is a kite. Explain below.
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4. Modify the vertices of the kite to make it convex (if necessary). Create the second diagonal of the kite and label the intersection of the diagonals F as shown in the diagram to the right.
5. Explain in a complete sentence below how the diagonals of a kite relate to each other. Make necessary measurements on the calculator to justify your answer below. Differentiate the diagonals by calling CD the symmetrical diagonal.
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6. Clear the screen. Construct another kite using a different method based on your explanation in step 5. Use an exploration from chapter five for additional assistance. Explain the steps to your construction below.
EXPLORATION 6-5. CONSTRUCTING A TRAPEZOID
Objective: Construct a trapezoid and an isosceles trapezoid, and explore their properties.
1. CREATE A NEW DOCUMENT with the variable trapez.
2. A trapezoid is a quadrilateral with at least one pair of parallel sides. Construct a trapezoid that is not a parallelogram. Explain the steps to your construction below. Make necessary measurements on the calculator to justify why this is a trapezoid and explain below.
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3. Measure and label a pair of consecutive angles between a pair of parallel sides, as shown in the diagram to the right. 4. Drag the endpoints of the original segment until you see the angles change measurement. What relationship exists between these angles? |
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5. Clear the screen. An isosceles trapezoid is a trapezoid that has a pair of base angles equal in measure. Construct an isosceles trapezoid using the Isosceles Trapezoid Symmetry Theorem and the drawing related to the theorem on page 330 of your text. Hint: Use a Reflection. Explain the steps to your construction below.
6. Measure the length of the legs, the non-parallel sides, of the isosceles trapezoid. Drag around the vertices until the measurements change. What relationship exists between the legs of an isosceles trapezoid?
EXPLORATION 6-M. MYSTERY QUADRILATERALS
You will need Graph Link cable and software to download
the files used in this exploration. Make sure that your browser
preferences are set to read graph link files with extension 92A.
Once you are ready, click on each of the file names to download:
| quad1.92A | quad2.92A | quad3.92A | quad4.92A | quad5.92A | quad6.92A |
Objective: Determine the most specific name for each mystery quadrilateral (parallelogram, rectangle, square, rhombus, trapezoid, isosceles trapezoid, kite, or quadrilateral).
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You must provide a valid "proof" for each mystery quadrilateral. For each quadrilateral, write a short paragraph (see below for an example). Support your answer with the measurements you have taken. Make sure that you list all relevant measurements on your lab. Also be sure to state why the quadrilateral could NOT be any other name. Before getting started, fill in your hierarchy to refer back to if needed! |
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1. Open QUAD1 in the MYSTERY folder.
MYSTERY QUADRILATERAL #1
| Solution Paragraph. |
sketch of diagram and relevant measurements!
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2. Open QUAD2 in the MYSTERY folder.
MYSTERY QUADRILATERAL #2
| Solution Paragraph. |
sketch of diagram and relevant measurements!
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3. Open QUAD3 in the MYSTERY folder.
MYSTERY QUADRILATERAL #3
| Solution Paragraph. |
sketch of diagram and relevant measurements!
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4. Open QUAD4 in the MYSTERY folder.
MYSTERY QUADRILATERAL #4
| Solution Paragraph. |
sketch of diagram and relevant measurements!
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5. Open QUAD5 in the MYSTERY folder.
MYSTERY QUADRILATERAL #5
| Solution Paragraph. |
sketch of diagram and relevant measurements!
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6. Open QUAD6 in the MYSTERY folder.
MYSTERY QUADRILATERAL #6
| Solution Paragraph. |
sketch of diagram and relevant measurements!
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EXPLORATION 6-6. ROTATION SYMMETRY
Objective: Compare the number of vertices in an object to its angle of rotation.
1. CREATE A NEW DOCUMENT with the variable rotsym.
| 2. Place point A in the middle of the screen. Create an arc, starting at point A, and ending near the top of the screen. Create and measure an angle in the lower left-hand side of the screen as shown in the picture to the right. |  |
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3. Rotate the arc about point A using the angle. Repeat the arc rotations until the arc has gone completely around (make seven arcs). Note that the arcs may not perfectly overlap. |
| 4. Create a seven sided polygon (heptagon) which has vertices touching the ends of each arc as shown in the diagram to the right. |  |
| 5. Adjust the angle in the lower-left hand side of the screen until there is exactly six spokes/vertices as shown above in the picture to the right (two arcs/vertices overlap). The arcs do not have to be perfect, but should be close to overlapping. Record the information below. |  |
| #arcs/vertices | name of polygon | degree of rotation |
| 6 | hexagon | |
| 5 | pentagon | |
| 4 | quadrilateral | |
| 3 | triangle |
6. Adjust the angle in the lower-left hand side of the screen until there are exactly five spokes/vertices. Record the information above. Repeat the procedure until the table above is complete.
7. The overlapping arcs/vertices occur in this exploration because you are creating a special type of polygon. Make up a name that you would use to classify these polygons.
Make necessary measurements on the calculator to justify why you have chosen this name. Explain below.
8. Write an equation below relating the number of vertices (V) in this special type of polygon compared to the degree rotation (D) between consecutive vertices in the polygon. Explain how you determined this.
EXPLORATION 6-7. REGULAR POLYGONS
Objective: Determine properties of a regular polygon
1. CREATE A NEW DOCUMENT with the variable regpoly.
| 2. Create a regular pentagon that will cover a large portion of the screen as shown to the right. Press F3 and choose 5: Regular Polygon, move the pencil to the center of the screen, press ENTER to locate the center, move out from this center (you'll see a dotted circle) as far as you can before part of the circle goes off the screen. Press ENTER. Move the pencil away from the point on the dotted circle until the screen reads {5} and press ENTER. You will see a regular pentagon as shown to the right. |  |
3. Measure the side lengths and angles of the regular pentagon. Explain below two properties of regular polygons which you discovered.
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4. Connect seven segments from the center of the pentagon to each of its vertices. Five triangles should form as shown in the diagram to the right. Do you know which company uses this design as their logo?
5. Without measuring on your calculator, determine the measure of each of the three angles in one of these triangles. Explain your reasoning and show your work below. Hint: Look back at Exploration 6-6. |
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6. Check your answer to step five by measuring each angle of one of the triangles. If your answers do not match, go back and check your work.
Last revised: August 2000