Ratios

Suppose we have a point X on a segment AB. This point divides the segment into two pieces: AX and XB. It is often important to know the ratio of the lengths of these two segments. this is sometimes referred to as the ratio into which X divides the segment AB. There are two common ways to measure this:

You can compute the ratio Length(AX)/Length(XB), which describes the relative sizes of the two pieces. If X is close to A this ratio is close to 0. As X moves along the segment towards B, the ratio gets increases. When X is at the midpoint of AB the ratio is exactly 1. As X gets closer and closer to B the ratio gets larger and larger, without bound.

 

You could also compute the ratio Length(AX)/Length(AB). This measures what fraction of the whole segment is taken up by AX. When X is close to A, this ratio is close to 0. Again, as X moves toward B the ratio increases. This time, however the ratio is 1/2 when X is at the midpoint and is 1 when X is at B.

 

Let us take some time to see how to display these ratios effectively in Sketchpad.

1. Open a new sketch. Use the Line tool to draw a horizontal, dashed line, then hide the two defining points for the line.

    

2. Use the segment tool to draw a segment with both endpoints on this line. Make this segment thick and blue. Label the endpoints A and B.

    

3. Construct a point on the segment AB and label it X. Then draw the segment AX and make it thick and red. At this stage your sketch should look like You should be able to slide X back and forth along AB and see the red segment AX expand and contract.

    

4. At this stage we can measure one of the ratios. We really don't need the dashed horizontal line any more so you can select and hide it. Then, holding the shift key down, select first the segment AX and then the segment AB. Choose Ratio from the Measure menu. You should see the ratio displayed. The first segment selected gives the numerator and the second the denominator so the order in which you select the segments is crucial. Make sure that you have the correct ratio displayed.

    

5. To get the first ratio we discussed, you need to construct the segment XB (make it thick and green. Then, holding down the shift key, select first the segment AX and then the segment XB. Note that clicking on either of these segments could actually select segment AB. If this happens, just click again and you should get the next alternative, which will be the segment you desire. Be sure to save this file. You will need it later in the lesson.

Dilations

A dilation is a transformation that contracts (or expands) all points towards (away from) a given central point by a fixed ratio. Thus, to specify a dilation you have to specify the central point and the ratio. If the ratio is less than 1 then points will be pulled in towards the central point and all distances will shrink by the ratio. In other words, if two points P and Q were distance 6 apart and you applied a dilation with center C and ratio 1/3 then P would go to a point P' and Q would go to a point Q' where now the distance from P' to Q' would be 2.

Open the first sample sketch for this lesson (Open file ex2_1.gsp) and experiment with the results of a dilation. This sketch has defined a dilation centered at C with ratio determined by AX/XB. The three blue figures are the originals and the results of dilating them are colored in magenta. Try moving X to get different ratios and moving C to give the original figures a different position relative to the center. You can also change the shape and size of the original figures.

You should be able to make some simple conjectures about the relationship of a figure to its image after a dilation. For instance:

• A circle always dialtes to a ......

   

• A segment always dilates to a .......

   

• A triangle always dilates to a ......

   

When you consider how you would complete these conjectures, fill in both the kind of figure (segment, circle, ellipse rectangle etc.), the shape (similar to the original, larger angles than the original etc.), and the size (relative to the original).

Now let's see if we can create an interesting dilation.

1. Open the sketch you saved from earlier in the lesson. Draw a segment at an angle across the sketch. Label the endpoints P and Q. We are going to construct a point Y on PQ so that the ratio PY/YQ is exactly the same as the ratio AX/XB. We will do this by using a dilation to "pull" Q down towards P.

    

   

    

2. Start by setting the center of the dilation at P: use the selection tool and double click on P. You should briefly see a sort of bulls-eye appear around P.

    

3. Select Q and then choose Dilate... from the Transform menu. A dialog box will appear. This lets you set the dilation ratio. It is clear where to enter the ratio you want. In our situation, however (as is often the case), we don't know what the ratio is. We want to use a quantity that is determined by the configuration to set the dilation ratio.

   Notice that the dialog box refers to the ratio in terms of the New distance and the Old distances. In other words, the ratio of PY to PQ. But this is what we want to be the same as the ratio of AX to AB. This is a quantity we have already measured.

    

4. Click on the measurement of the ratio AX/AB in the sketch. The dialog box should change to say that you are now dilating by a Marked Ratio. Click OK. You should see a new point on the segment PQ. Label this point Y.

    

5. Slide X along AB and observe Y slide along PQ. Change the slope and length of PQ and slide X some more.

    

Save this sketch. You will need it in your homework assignment.

Homework

Complete the exercises in Lesson 2 homework and submit them to the MTL handin system.