GEOMETER SKETCHPAD INTRODUCTION

Transcription

1 GEOMETER SKETHPD INTRODUTION ctivity 1: onstruct, Don t Draw onstruct a right triangle Use the line segment tool, and draw a right angled triangle. When finished, use the select tool to drag point to the left. Triangle is no longer a right angled triangle. This is called drawing a right angled triangle. When you draw an object, it will not retain the geometric properties if one of the vertices is moved. omplete the following instructions to construct a right angled triangle. If you move any of the vertices, it will remain a right triangle. Start a new sketch. Draw a vertical line segment. Select the line segment and a point which is on the line segment, and construct a perpendicular line. (onstruct Perpendicular Line) Geometer will construct a line perpendicular to the selected line and through the selected point. reate a point, on the vertical line. Draw a line segment from point to the line forming a right angled triangle. all the last vertex of the triangle point. Try dragging any of the points, or to distort the right angled triangle. will always be a right triangle!!! Trick: select the line through and hide it (Display menu or use ommand H); then draw a line segment through creating a triangle guaranteed to be right. onstruct an equilateral triangle Draw any line segment. Select point, and under the Transform menu, mark centre. This marks point as the centre of rotation. Select the line segment and point, and using the Transform menu, rotate by 60 degrees. (counterclockwise is positive 60º, clockwise is -60º) Try to complete the equilateral triangle by marking a point as centre, then rotating a point and line by 60º. Drag any of the three points to distort, but it will always be and equilateral triangle. Practise onstruct a square, rectangle, and a regular (i.e. all sides and angles equal) pentagon.

2 ctivity 2: Helpful Hints onstructing Lines and Segments Place two points anywhere on the page. Select both points and click the onstruct menu. If line segment is selected, the onstruct menu will create a line segment. If ray is selected, the onstruct menu will create a ray (select the points in order). If line is selected, the onstruct menu will create a line. Explore some features (think about the following questions) 1. Why are some commands grey and some are black? 2. How do you know that an object has been selected? 3. How do you turn the label of an object on and off? How do you rename a label? How do you move labels? 4. Using the selection tool, double click on a point. What does this do? 5. Using the selection tool, double click on a line. What does this do? Does it matter if you double click on a line segment or a ray? 6. How do you change the font and size of the labels and text? 7. How do you measure an angle? 8. Does Sketchpad measure reflex angles? Practise hange the font type and sizes in a label Show measurements for the side of the square Show the angle measures for the pentagon

3 ctivity 3: The Sum of the ngles in a Triangle (STT) Hint: If a Measure menu feature is greyed out (not available), check if you have selected the appropriate ingredients. For example, you cannot measure an angle unless exactly 3 points are selected. Measuring ngles onstruct an arbitrary triangle and label the vertices, and Select vertices, and in that order Now measure the angle (using the Measure menu). In the same way, measure angle and angle. Use the calculator (Measure alculate ) to find the sum of the three angles. Do not type in the actual measurements; simply highlight each of the three angle measurements with plus signs in between. m = 29 m = 80 m = 71 m + m + m = 180 Making a table Select all three angle measurements and the angle sum. onstruct the table by selecting tabulate from the Number menu. To add numbers to your table, change the current triangle by dragging a side or vertex to a new position. Double-click on the table using the pointer tool. The new values should be displayed as a new column in the table. Make four more entries in your table. (Note that you can move the entire table to a different location in your sketch by clicking and dragging.) Your table might look something like this one: m m m m +m +m Question: When you change the shape of the triangle what happens to the angle measurements and what happens to the angle sum?

4 ctivity 4: The Triangle Inequality Open a new sketch and construct an arbitrary triangle. Label the vertices of the triangle. (You can move the vertex labels so that they are not on top of the vertices - use the finger tool plus click and hold.) If the vertices are not labeled, and, relabel them by double-clicking the finger tool. This will allow a label change. Select side. Using the Measure menu, determine the length of side and the length of side. Use the calculator to find the sum of the lengths of sides and. Now measure the length of side. ompare the sum of the lengths of and with the length of. reate a table as you did in Lab 3 to compare the length of and the sum of the lengths of and. (include at least 4 rows in your table) Questions: omplete the sentence: The sum of the lengths of two sides of a triangle is the third side. This relationship is the triangle inequality. Move vertex closer to side. How does this affect the comparison between the sum of the lengths of two sides and the length of the third side? What will happen to the triangle inequality if you move vertex onto side? In your opinion, if vertex lies on segment is the figure still a triangle?

5 ctivity 5: Exterior ngles D Exterior ngles of a Triangle onstruct where the base lies on a ray as shown. Vertices and should be the control points of the ray. onstruct a point D on the ray. The angle D is an exterior angle of the. Measure angle D. ngles and are in the interior of the triangle and are on the opposite side of the triangle from angle D. These angles are called interior opposite angles. Measure angle and angle then use the calculator to find their sum. Make a table showing the value of angle D, the values of angles and as well as the sum of these interior angles. Now add four more rows of entries to your table. Question: What do you notice about the relation between the exterior angle and the sum of the two interior opposite angles? Sum of the Exterior ngles of a Triangle Use three rays to construct a triangle as shown. (Remember to use the control points as vertices.) Measure all three exterior angles and find their sum. reate a table using values for several different triangles. Question: What do you notice about the sum of the exterior angles of a triangle? Sum of the Exterior ngles of a Quadrilateral Now construct any quadrilateral using four rays. Measure the four exterior angles and find their sum. reate a table using results for several different quadrilaterals. Question: What is your conclusion?

6 Sum of the Exterior ngles of a Polygon Use Sketchpad to find the sum of the exterior angles of a pentagon. Questions: What do you think is true of the sum of the exterior angles of any polygon? Test your answer for a dodecagon. ( dodecagon is a polygon having 12 sides.) an you think of a way to prove your conclusion for all polygons? oncave Polygons reate a sketch for the exterior angles of a heptagon. Measure the exterior angles and their sum as before. Now drag one of the vertices toward the centre of the heptagon until it looks like the one below. (This heptagon which goes in on itself is called a concave heptagon.) Questions: What has happened to the sum of the exterior angles? an you think of an explanation for this? (Hint: It has to do with the way Sketchpad measures angles)

7 ctivity 6: Interior ngles Interior ngles of a Quadrilateral Now construct a quadrilateral. ecause of Sketchpad's inability to deal with reflex angles, do not construct a concave quadrilateral. Measure the four interior angles and find their sum. reate a table using results for several different quadrilaterals. What is your conclusion? Interior ngles of Polygons reate a pentagon, making sure that it is not concave. Measure the interior angles and find their sum. Drag one or more of the vertices. What do you notice about the sum of the interior angles of a pentagon? Repeat this for different types of polygons. omplete the following chart: POLYGON NUMER OF SIDES SUM OF THE INTERIOR NGLES SUM OF THE EXTERIOR NGLES Triangle Quadrilateral Pentagon 6 7 Octagon Nonagon gon 100 n-gon n

8 ctivity 7: Pythagorean Theorem Verifying the Pythagorean Theorem The sum of the squares of a right angled triangle is equal to the sum of the squares of the other two sides; or, is a right triangle, where = 90º, if and only if a b c. c^2 c a a^2 rea a^2 = cm 2 b rea b^2 = cm 2 ( rea a^2 ) + ( rea b^2 ) = cm 2 rea c^2 = cm 2 onstruct a right angled triangle labeled, where is the right angle label the sides a, b and c b^2 construct squares from each side by rotating line segments (remember positive angles are counter clockwise, negative angles are clockwise) select the 4 points surrounding each square, and construct the interior; repeat for the other 2 squares select the area, and measure the area of the square; repeat for the other 2 squares right click on the largest area measurement and select Properties Label the measurement c^2. Repeat for the other two area measurements, labeling them a^2 and b^2. Using the calculate feature, demonstrate the Pythagorean Theorem, by showing that a^2 + b^2 = c^2 colour the squares (optional) The onverse of the Pythagorean Theorem Show that the Pythagorean Theorem only works for right angled triangles. Draw an arbitrary triangle. using rotated line segments, construct a square at each side of the triangle. construct an interior of each of the squares. determine the areas of each of the squares label the measurements of the areas of the squares a^2, b^2 and c^2 measure Move any of the points, or and verify that a b c will be true only if = 90º.