Solutions for Session 8, Part A

See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6| A7 | A8

 Problem A1 Picture #3 seems to be the most faithful replica of the original picture because proportions between the various body parts are best preserved.

 Problem A2 The two rectangles are similar. Their angles are the same and equal to 90°. To check whether the sides are in proportion, first set up the following ratios of the corresponding sides: If the two are equal, the sides are in proportion. So we have 2/6 = 3/9, and they are both equal to 1/3. So we can say that the corresponding sides are 1:3 ratio.

 Problem A3 The two triangles are similar. By measuring pairs of sides, we can see that they are proportional. In addition, we can measure pairs of corresponding angles and see that they are of equal measure.

Problem A4

 a. The coordinates of the figure are as follows: A = (0,2) B = (0.5,3) C = (1,2) D = (2,2) E = (2.5,3) F = (3,2) G = (2,0) H = (1,0) b. The coordinates of the new figure are as follows: A' = (0,4) B' = (1,6) C' = (2,4) D' = (4,4) E' = (5,6) F' = (6,4) G' = (4,0) H' = (2,0) c. Follow the instructions -- see figure below. d. The two figures are similar. We can see this by checking that the corresponding angles have the same measure. The corresponding sides are in a 1:2 proportion by construction.

 Problem A5 If you changed the angles, you wouldn't have the same kind of shape anymore. For example, in a square, all of the angles are 90°. If you double them all, you will simply have a straight line (a bunch of 180° angles lined up), which is certainly not "similar" to a square. If we start with a triangle, the angle sum is 180°. If we double all of the angles, we would end up with something with angles summed to 360° -- certainly not a triangle, and not similar to what we started with.

 Problem A6 The distance between the two points is .

 Problem A7 The new coordinates are (2b1,2b2) and (2a1,2a2). The new distance is . The new distance is double the original distance. We can show this by factoring the 22 from the second expression, obtaining the following: . So the new distance is exactly twice the original distance.

 Problem A8 We would need to find at least two well-defined reference points on the figure (such as the two endpoints of the spirals). In relation to these two points and the line they define, we could establish pairs of corresponding points on the two shapes. We would then require that all of the distances between all pairs of corresponding points be proportional.