Geometry Activity: Fermat s Point Casio Classpad 300 vs. TI-89 CALCULATORS: Casio: ClassPad 300 Texas Instruments: TI-89, TI-89 Titanium Fermat s Point Fermat s point is the point such that the sum of the distances from this point to the vertices of a triangle is minimal. The applications of Fermat s point are numerous. For example, if a company had three stores and only one warehouse, to minimize the distances from the warehouse to the stores, it should locate the warehouse at Fermat s point for the triangle formed by the three stores. Here s how to use the ClassPad 300 to find Fermat s Point. Using the Casio ClassPad 300 C.C. Edwards Start a geometry session: Use the stylus to tap Menu on the Icon Panel. Tap Geometry to start a geometry session. Start a new file: Tap File on the Menu bar. Tap New in the drop down menu. Tap OK to clear the screen and start a new file. Construct the triangle: Tap the Triangle tool in the Shape menu (third drop down menu). Triangle Tool 1
Using the Casio ClassPad 300 Fermat s Point (continued) Place the stylus near the top middle of the screen and drag it to the middle right of the screen, as illustrated in the figure at the right. When you lift the stylus the triangle appears. Anchor the vertices of the triangle: Tap the arrow at the right end of the Toolbar to access the Measurement tool. To anchor each vertex: Tap the vertex to display its coordinates in the Measurement box. If you d like to change the location of that vertex, drag the stylus over the entry in the Measurement box and then use the keypad to enter the coordinates of the desired location. Tap the Check box R to the right of the Measurement box to anchor the vertex. Then tap a blank area of the screen to deselect the vertex. Tap the left arrow at the right end of the Measurement tool to return to the Toolbar. Approximate the location of Fermat s Point and then construct segments joining this point to the vertices of the triangle Tap the Point tool in the Draw menu (second drop down menu). Tap the screen in the general location of where you think Fermat s point may be located. Tap the Segment tool in the Draw menu. Join each vertex to "Fermat s point" by tapping the vertex and then tapping "Fermat s point," as illustrated in the figure at the right. 2
Using the Casio ClassPad 300 Fermat s Point (continued) Rotate segments AD and DC 60 o about vertex C: Tap the Selection tool on the label of the View menu (first drop down menu). Tap segment AD to select it. Then tap segment DC. Tap the Rotation tool in the Construct menu (fourth drop down menu). The directions at the bottom of the screen tell you to tap the rotation center. Tap vertex C. Use the keypad to enter 60 and then tap OK. This rotates segments AD and DC 60 o about vertex C, as illustrated in the figure above. Construct segments D D and A B: Tap a blank area of the screen to deselect segments AD and DC. Tap the Segment tool on the label of the Draw menu. Tap point D and then tap point D. Tap point A and then tap point B. Since CD was rotated 60 o to CD, CCD is an equilateral triangle. So m(d D) = m(cd). Thus the length of the broken path A D + DD + DB is the same as the sum of the distances from D to the vertices of the triangle. Since the shortest distance between two points is a straight line, the path A B is shorter than path A D + DD + DB. So to find Fermat s Point, we need to move point D so that the broken path A D + DD + DB and the shortest path A B coincide. Hide segment BD: (This makes it easier to select segment A B.) Tap the Selection tool on the label of the View menu. Tap segment BD to select it. 3
Using the Casio ClassPad 300 Fermat s Point (continued) Tap Edit on the Menu bar. Tap Properties in the Edit menu. Tap Hide, as illustrated in the figure at the right. Anchor point D on segment A B: Tap segment A B and then tap point D. Tap the right arrow at the right end of the Toolbar to access the Measurement tool. Tap the Incidence tool in the drop down menu to the left of the measurement box, as illustrated in the figure at the right. "No" in the Measurement box indicates that D is not on segment A B. Tap the Check box to anchor point D on the segment A B. Construct Fermat s point: Tap a blank area of the screen to deselect all selected items. Tap point D and then tap segment A B. Tap the Check box to anchor point D on the segment A B. Tap a blank area of the screen to deselect the point and the segment. Tap point D. The coordinates in the Measurement box are the coordinates of Fermat s Point. 4
Geometry Activity: Fermat s Point Casio Classpad 300 vs. TI-89 CALCULATORS: Casio: ClassPad 300 Texas Instruments: TI-89, TI-89 Titanium Fermat s Point Using Geometer s Sketchpad on a TI-89 Titanium Fermat s point is the point such that the sum of the distances from this point to the vertices of a triangle is minimal. The applications of Fermat s point are numerous. For example, if a company had three stores and only one warehouse, to minimize the distances from the warehouse to the stores, it should locate the warehouse at Fermat s point for the triangle formed by the three stores. C.C. Edwards Here s how you solve this problem using Geometer s Sketchpad on a TI-89 Titanium. Install the Sketchpad application on your calculator: Sketchpad isn t preloaded on the TI-89, so you ll have to download it from Texas Instruments website and install it on your calculator. The Sketchpad application is free and the cable connection the TI-89 to your PC came with your calculator. The cable linking a TI-89 to your PC needs to be purchased separately. Start Sketchpad and, if necessary, clear the screen: Press APPS. Repeatedly press the Arrow keys to highlight the Sketchpad application, as illustrated in the figure at the right. Press ENTER to start Sketchpad. Press F1 to view the options in the Sketch menu. Press to tell the calculator that you want to start a new file. If you are asked if you want to save any previously existing sketch, press ESC to clear the screen. 5
Using Geometer s Sketchpad on a TI-89 Titanium Fermat s Point (continued) Turn labels on: Press F1 ENTER to access the Preferences menu. Press to display the options for labeling points. Press 2 to tell the calculator to label constructed points. Press ENTER to save your changes. Construct the triangle: Press 2nd F3 ENTER to activate the Segment tool. Use the Arrow keys to move the cursor to the location of one vertex of the triangle and then press ENTER. Use the Arrow keys to move the cursor to the location of another vertex of the triangle and then press ENTER. Press ENTER to start a new segment. Use the Arrow keys to move the cursor to the location of third vertex of the triangle and then press ENTER. Press ENTER to start the third side of the triangle. Use the Arrow keys to move the cursor to the first vertex of the triangle and then press ENTER. Press ESC to deactivate the Segment tool. Press ESC again to deselect all items. 6
Using Geometer s Sketchpad on a TI-89 Titanium Fermat s Point (continued) Approximate the location of Fermat s Point and then construct segments joining this point to the vertices of the triangle: Press 2nd F3 to activate the Point Tool. Use the Arrow keys to move the cursor to the approximate location of Fermat s point and press ENTER to construct that point. Press 2nd F3 ENTER to activate the Segment tool. Use the Arrow keys to move the cursor to one vertex of the triangle and then press ENTER. Use the Arrow keys to move the cursor to Fermat s point and then press ENTER. Repeat Steps 4 and 5 to connect Fermat s point to the other two vertices. Press ESC to deactivate the Segment tool. Press ESC again to deselect all items. Rotate segments AD and DC 60 o about vertex C: Use the Arrow keys to move the cursor to vertex C and then press ENTER to select that point. Press F4 1 to mark C as the center of rotation. Press ESC to deselect C. Use the Arrow keys to move the cursor to segment CD and then press ENTER to select that segment. Use the Arrow keys to move the cursor to point D and then press ENTER to select that point. Use the Arrow keys to move the cursor to segment AD and then press ENTER to select that segment. Use the Arrow keys to move the cursor to point A and then press ENTER to select that point. Press F4 8 to activate the Rotation tool. Press 6 0 to set the angle of rotation to 60 o. Press ENTER to rotate the selected items 60 o about point C. Press ESC to deselect all items. 7
Using Geometer s Sketchpad on a TI-89 Titanium Fermat s Point (continued) Construct segment D D: Press 2nd F3 ENTER to activate the Segment tool. Use the Arrow keys to move the cursor to D and then press ENTER. Use the Arrow keys to move the cursor to D and then press ENTER. Since CD was rotated 60 o to CD, CCD is an equilateral triangle. So m(d D) = m(cd). Thus the length of the broken path A D + DD + DB is the same as the sum of the distances from D to the vertices of the triangle. Construct segment A B: Use the Arrow keys to move the cursor to A and then press ENTER. Use the Arrow keys to move the cursor to B and then press ENTER. Press ESC to deactivate the Segment tool. Press ESC again to deselect all items. Since the shortest distance between two points is a straight line, the path A B is shorter than path A D + DD + DB. So to find Fermat s Point, we need to move point D so that the broken path A D + DD + DB and the shortest path A B coincide. Find Fermat s point: Use the Arrow keys to move the cursor to point D and press ENTER to select it. While holding down the ALPHA key, use the Arrow keys to move D to segment A B. Again while holding down the ALPHA key, use the Arrow keys to move D along segment A B until the broken path A D + D D + DB coincides with segment A B. Press F5 ENTER to display the coordinates of Fermat s point. Press ESC to deselect all items and then use the Arrow keys to move the cursor away from the figure. 8
The Casio Advantage... Geometry software is preloaded on the Classpad 300 No extra steps needed. Pen touch for easy manipulation Large screen makes it easy to see The ClassPad 300, unlike Sketchpad on a TI-89 or Voyage 200, has the ability to move and anchor a point on a line. 9