This lab consists of two main parts. The first part is about coordinates and about rotations. It is an important warm-up for the class that Leah will lead on Friday. The second part is a mixture of Sketchpad and physical models that extends some of the work we have done with conics.

In this part of the lab we will take a blank plane and create a system of coordinates. We will ignore for the moment the built-in coordinate system that comes with Sketchpad. Here are the steps.

• Draw a line AB and construct a circle with center A through B. Construct the line through A perpendicular to line AB.
• With the point tool click in a point C which is a point of intersection of the perpendicular line and the circle. There are two points of intersection; choose the point C so that C is the rotation of B by 90 degrees with center A.
• Relabel points A, B, C to be points O, E1, E2.
• Now draw a point P anywhere in the plane and drop perpendiculars to the two lines. In other words, construct the two lines through P with are perpendicular to the lines OE1 and OE2. Construct the points of intersection that are the feet of the perpendiculars and label the point on OE1 as P1 and the other as P2. You may wish to color this new lines or make them dashed to distinguish them.

Now we consider the point O as the origin and the line OE1 the x-axis, with |OE1| as the unit length. Then we can have Sketchpad compute the coordinates of P by computing ratios as follows.

• Select the 3 points (IN ORDER) O, E1, P1 and HOLD DOWN THE SHIFT KEY while you select Ratio from the Measure menu. This will measure the ratio OP1/OE1 (but the order of points determines which ratio you compute). This is the x-coordinate.

• In the same way, select the points O, E2, P2 and measure the ratio. This is the y-coordinate.

Finally, we can perform a little trick and label the coordinates differently. To do this, we want to make a second copy of the ratios. A simple way is this.

• Bring up the Calculator and then just click on a ratio measurement and then click on OK, without performing any calculations. Do this for each ratio. You now have an original and copy of each ratio.
• Now, double-click on the COPY ratio OP1/OE1 WITH THE FINGER (TEXT) TOOL. Then click the radio button that says Text Format. The text OP1/OE1 will be selected; WITHOUT clicking just type x to replace the selected text so that it reads x = … as in the figure. Then click OK to close the dialog.
• Use the same method on the other ratio to get y = ….

Finally, Select All and choose Make Script from the Work menu. This script should have 3 givens. It can be used to construct coordinate systems in one step. Be sure to select the ratios measures as well as the geometric objects. You want it all.

Now start with a new sketch. This time we will use the built-in coordinates.

• In a new sketch, choose Create Axes from the Graph menu. This will create the origin A and the unit point B.
• Draw a point P anywhere in the plane.
• Apply your script to points A, B and P to get our do-it-yourself coordinates and points P1 and P2 and E2. Construct the interior of the rectangle AP1PP2 and shade it a light shade.
• Also Measure the built-in coordinates of P and see that you get the same result.
• Now draw a new line AX through A. Construct point F1 with the point tool as the point of intersection of the unit circle and the line, choosing the point of intersection on the same side of A as X.
• First we measure the coordinates of this point F1 using the Measure menu. Next we wish to compute the coordinates by using the angle BAX. Before measuring this angle, go to Preferences and choose Directed Degrees as the unit of measure of angles. Then measure angle BAX and use the calculator to compute the cosine and sine of this angle. Compare these numbers with the coordinates of F1. Explain.

We can think of this line AX as the image of the x-axis by a rotation.

• Choose the points B, A, X (IN ORDER) and choose Mark Angle from the Transform menu. Select point A and choose Mark Center from the same menu.
• Now select points P, P1, and P2 and also E2 and the rectangle interior and choose Rotate by Marked Angle. Color the rotated rectangle a different color.
• Observe that if the coordinates of P', the rotated P, with respect to the rotated coordinate system, are the same as the coordinates of P in the original system. Tell which ratios these coordinates are.
• Multiply the cosine and sine calculated above by x. Also, measure the coordinates of P1'. Explain why these are the same numbers. How are the coordinates for P2' computed from x, y and the cosine and sine of the angle BAX?

Now we turn things upside down a bit.

• Start with a new sketch and Create Axes and then draw P and compute x and y as before.
• Also draw AX as before. Construct through A the perpendicular AY to AX and let F1 be constructed as above. Construct F2 as the intersection of line AY and the unit circle that is the 90-degree counterclockwise rotation of F1.
• Select the measure x and choose Mark Scale Factor in the Transform menu. Select A and choose Mark Center. Now, select point F1 and choose Dilate by Marked Scale Factor to get F1'.
• Repeat this with y. Mark y as the scale factor and dilate F2 by marked ratio to get F2'.
• Now we construct P' as a vector sum or by assuming that P1' = F1' and P2' = F2'. Construct the line through F1' perpendicular to AX and the line through F2' parallel to AX. Let the intersection be called P'.
• Construct the rectangle AP1PP2 and also the rectangle AF1'P'F2' to see better how we have constructed the rotation of the rectangle without using any rotation or reflection.