Construct (with Sketchpad or straightedge and compass) the figures that call for construction.
(a) Let F(x, y) = (3x – 2, 3y – 10). Draw a triangle ABC on graph paper and then draw the image of the triangle A'B'C' under F. Draw the lines (not segments) AA', BB', CC'. Do they appear to be concurrent? Tell what appears to be the center and what is the ratio.
(b) Investigate F algebraically. Find the center of F by solving (a, b) = F(a, b). Use the fixed point to regroup the formula for F so that it is clearly a dilation. Hint: If (a,b) is the center, use new variables u = x - a and v = y - b.
On graph paper, draw a line through (0,0) with nonzero slope. Pick two points on the line and draw the feet of the perpendiculars to the x-axis and y-axis. Show how the proportional relations in the figure lead to the equation of the line in terms of x and y.
Definition: Two figures have a center of similitude A if there is a dilation with center A that takes one figure to the other. Some figures have more than one such center.
In each case, draw the figures and construct the centers of any dilations that take one of the two figures to the other.
(a) Two parallel segments of different length.
(b) Two circles of different size.
(c) A parallelogram and itself.
In the next two, work with a "random" triangle ABC and its midpoints A'B'C'.
(d) Find the centers of similitude of triangle ABC and B'A'C.
(e) The same triangle ABC and triangle A'B'C'.
Draw two parallel segments AB and CD of unequal length. Construct the two centers of similitude for the two segments as in Figures 10 and 7 of EST. Then answer the following questions.
(a) Write a clear proof, using the idea of dilations, that midpoints of the two segments are collinear with the two centers of similitude. Hint: Prove that a dilation takes the midpoint of a segment AB to the midpoint of the image segment A'B'.
(b) Interpret (a) as a statement about trapezoids that we encountered as an earlier problem
Draw two circles with different radii in Sketchpad. Construct the two centers of similitude, E and I.
(a) Move the circles so that each is exterior to the other. Construct all common tangents. Print out the figure.
(b) Move the circles so that they are tangent to each other. Describe what happens to E and I and what happens to the common tangents.
(c) Move the circles so that they intersect in two points. Describe what happens to E and I and what happens to the common tangents.
(d) Move the circles so one is inside the other. Describe what happens to E and I and what happens to the common tangents.
(e) If O1 and O2 are the centers of the circles, prove that EO1/EO2 = - IO1/IO2.
With straightedge and compass or with Sketchpad, draw two intersecting lines and a point P as in Example 2 of EST. Then construct the two circles through P that are tangent to both the lines.