Lab 10 (12/5/01): Dilations

How to Dilate by a fixed Ratio

A. Ratio equal 1/2

In a new sketch, draw a triangle ABC and also a point D. Select D and Mark D as Center (Transform Menu). Then Select the triangle ABC and choose Dilate from the Transform Menu. You will have the opportunity to type in a ratio. Enter the ratio 1 to 2. This will create triangle A'B'C'.

Now drag the points ABC around to convince yourself the two triangles are similar. But the relationship is stronger than that.
What segments are parallel? How are vectors AB and A'B' related?
Drag D so that it is superimposed on one of the vertices, such as A. What triangle is A'B'C' in this case?
Construct the lines DA, DB, and DC. Where are the points A', B', and C'?
Select the triangle A'B'C' and construct Hide/Show buttons and hide the triangle.

B. Ratio equal –1/2

Now continue with the same sketch, but this time select the triangle ABC and dilate by the ratio –1 to 2. This gives a new triangle, also called A'B'C' by Sketchpad.

Again, which segments are parallel? How are vectors AB and A'B' related?
For this new triangle A'B'C', how are the points arranged on the lines DA, DB, and DC?
Drag D so that A'B'C' becomes the midpoint triangle. Observe that D is located at the centroid of the triangle. How does this relate to the ratio 2/3 in the theory of concurrent medians?

Select the triangle A'B'C' and construct Hide/Show buttons and hide the triangle.

C. Ratio equal –1

Now continue with the same sketch. Again select the triangle ABC and dilate by the ratio –1 to 1. This gives yet another new a new triangle, also called A'B'C' by Sketchpad.

Again, which segments are parallel? How are vectors AB and A'B' related?
For this new triangle A'B'C', how are the points arranged on the lines DA, DB, and DC?
Move D to the midpoint of segment BC. What figure is formed.
What are the symmetries of the figure made up of triangles ABC and A'B'C'? Is there a difference between the transformations "dilation by ratio –1 with center A" and "point reflection with center A" and "rotation by 180 degrees with center A"?

How to dilate by a Marked Ratio

C. Ratio defined by 3 points

Draw a polygon, say a quadrilateral ABCD. Also draw a point O. Construct the line OA and a point A' on this line.

Now do this carefully. Select, IN ORDER, points O, A, and A'. Then HOLDING DOWN THE SHIFT KEY, select Mark Ratio from the Transform menu.

Also, holding down the shift key with the same points selected, in the same order, choose Ratio from the Measure menu. This will show the ratio that you have marked on the screen. It is not necessary but is informative.

Now drag A' up and down the line so that you understand what ratio you have measured. In particular, what is the ratio when A' is located at O, at A, at the midpoint of OA. When is the ratio –1? When is the ratio 2?

Now, you can use the marked ratio to dilate the quadrilateral. First, Select O and Mark O as Center. Then select the whole polygon and choose Dilate with the option By Marked Ratio that will come up in the dialog box.

D. Solve a problem by Dilation

Suppose you draw any triangle ABC and you want to draw inside the triangle a square, so that one side of the square is on BC and the other two vertices are on the other two sides of the triangle, like this figure:
Method: Start by constructing a "partial solution" to which you can add a parallel line to make a "scale model". Then dilate to get the solution. You will need to figure out the center of the dilation and mark the ratio in the sketch so that if you drag A, B, or C, the square will follow.

E. Given two circles, construct the centers of dilation and the common tangents.

Do this. This is problem 10-5. It is also described in the EST (Eye for Similarity Transformation) handout.

F. Construct a Circle given 2 tangents and a point on the circle

Do this. This is problem 10-6. It is also described in the EST (Eye for Similarity Transformation) handout.

G. Euler Line

Return to the first exercise. Begin with a triangle ABC and a point D. Dilate ABC by ratio –1/2.

Now in ABC, construct the altitudes and the orthocenter H. Also construct the perpendicular bisectors of the sides and use them to construct the circumcenter O and the circumcircle c. Use different colors for the altitudes and the perpendicular bisectors.

Now select the whole figure of ABC and its lines and circle and again dilate with center D and ratio –1/2. Observe that now you have the orthocenter, circumcenter, etc., in the smaller triangle A'B'C'. Also notice that D, H and H' are collinear and also D, O and O' are collinear. What are the ratios DH'/DH and DO'/DO?

Now drag D so that the triangle A'B'C' is superimposed on the midpoint triangle of ABC. Notice that this makes H' coincide with O (why?). Explain why, when H' = O, that D coincides with the centroid G.

Finally, explain why the points G, H, O and O' are collinear. Then using what you know about the ratios, explain how the 4 points are arranged. What are the endpoints of the smallest segment containing the four points? Where are the other 2 points located?

Finally, when D = G and H' = O, observe the circle c'. Since A', B' and C' are the midpoints of ABC in this case, the circle passes through the midpoint. But notice that it also passes through the altitudes in some special points. What are those points?