An email question.

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Morris Needleman recently asked the list:

"In GSP I have two points A and B. I wish to construct the locus of all points P so that PA=2*PB Any ideas?"

In this article are some worksheets that I have developed based on a response to the list by Anthony Harradine, Director of the Noel Baker Centre for School Mathematics. I have used Cabri instead of GSP because I think a 3D solution to this problem is important.

You do have permission to copy these two pages and cut out the three worksheets for student use. Eventually I will make the worksheets and finished browser models available on a new website which is under construction.

There is still some debate about an appropriate pedagogy when using this type of software. Recently I overheard a group of students complaining because their mathematics teacher built all the Cabri models himself and then presented his classes with the finished models that they were allowed to manipulate for a few minutes before heading back to their classrooms. At the other extreme are teachers who pose a problem and offer little further help.

These worksheets are designed to offer a compromise between these extremes. Some will find that the worksheets spoon-feed the students too much while others will take the ideas offered and add much more detailed instructions.

Some will argue that it is important that students are able to use the software to build their own models, and if they follow these worksheets, they are still only building our model, not theirs. Others use worksheets like these to offer students, and teachers, examples which they can use to build their skills before flying solo.

Exercise 1

Given two points A and B. Construct the locus of all points P such that PA = 2 x PB

A discovery approach.

Open Cabri Geometry and locate the menu items shown in bold font in the text below.

Place two points, A and B (see diagram below).

Label the points.

Draw the line segment AB. Find the length of AB.

Carefully adjust the position of B until the line segment AB is horizontal and 6 cm long.

Place a point P and draw segments AP and BP.

Find the length of the segments AP and BP.

[ILLUSTRATION OMITTED]

Adjust the position of P until AP is about twice as long as BP as shown in the diagram below.

The next bit is tricky. We want to move the point P but leave a new point where it used to be. If you try to put a new point on top of P, it will just move when you move P. If you try to make a new point near P the software will want to put it with P. What you need to do is keep a finger at the right spot on the screen, move P completely away and then put the new point next to your finger.

Now move P to another point where AP is twice as long as BP and repeat the same procedure to put a new point at this position.

Keep doing this until you can see the overall shape of the locus.

Now switch on trace and click on the point P. Move P as steadily as you can from one point to the other, always trying to keep the length of AP twice as long as the length of BP. If you want to start the trace again, just resize the window and the trace will be erased.

What shape did you get for your locus?

Exercise 2

Open Cabri Geometry and locate the menu items shown in bold font in the text below.

Place points A and O (see diagram below).

Draw the line segment AO. Find the length of AO.

Carefully adjust the position of O until the line segment AO is horizontal and 8 cm long.

Find C, the midpoint of AO.

Find B, the midpoint of CO.

Place points K and L. Find M, the midpoint of KL.

Draw a circle centre K through M.

Place the point N on the circle.

Join the line segments KL and KN.

Check what happens as you move the point L to the left and to the right. What is the relationship between the length KL and the length KN?

[ILLUSTRATION OMITTED]

Select the compass command. Click on the point A and then on the segment KL to draw a circle.

Repeat using the point B and the segment KN.

Find the points of intersection of the two circles.

Label these points P1 and P2.

What is the relationship between the length of AP1 and BP1? Check by finding their lengths.

Move the point L left and right.

What is the locus of points P1 and P2?

Turn on trace and click on points P1 and P2.

Move the point L left and right. What shape do you get? Check your answer by drawing a circle with centre O and passing through C.

Exercise 3

Open Cabri 3D and locate the menu items shown in bold font in the text below.

You may choose to Display using Scale 2.

Construct a vertical plane by clicking on the y and z unit vectors at the centre of the screen. Then hide the unit vectors.

To hide a construct, select it by clicking on it and then, if you are using a Macintosh hold down the command key and type M or, if using Windows, hold down the control key and type M.

Construct a line at the intersection of the planes.

Place a point at the left end of the line and type A.

Place another point, O, near the centre of the line.

Construct C, the midpoint of AO.

Construct B, the midpoint of CO.

[ILLUSTRATION OMITTED]

Place points K and L on the plane as shown.

Construct M, the midpoint of KL.

Construct the normal to the vertical plane at K by using the perpendicular command.

Construct a circle centred on the normal and passing through the point M.

Place point N on the circle.

Join segments KL and KN.

Check what happens as you move the point L to the left and to the right. What is the relationship between the length KL and the length KN?

Tidy up by hiding the planes, and the normal.

Draw a sphere with centre A and radius KL.

Draw a sphere with centre B and radius KN.

Construct the intersection curve of the two spheres by clicking one sphere and then the other.

Select any point P on the intersection curve.

Join the segments AP and BP.

Move the point L left and right.

[ILLUSTRATION OMITTED]

There is not yet a trace command in Cabri 3D but if you draw a sphere, centre O through C, you can see how closely the intersection curve hugs the new sphere as you move the point L.