Exercise 37 Page 342

Luke says that 23AP=AD. Let's think of whether he can be correct. We will use the diagram below.

As we can see, AP is a shorter segment than AD. Moreover, multiplying its length by a fraction 23, the measure becomes even less. Thus, it is not possible for 23AP to be equal AD. Luke is incorrect. Let's try to find the correct relationship between the lengths of these segments. On the diagram we are given three pairs of congruent segments. ABand BC CDand DE AFand FE This means that B, D, and F are the midpoints of AC, CE, and AE respectively. This allows us to conclude that EB, AD, and CF are the medians of the triangle △ ACE. A point of intersection of triangle medians is called a centroid. In our case, it is point P. Now we can use the Centroid Theorem. The medians of a triangle inersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side. According to this theorem, centroid P is situated two thirds of the distance from vertex A to the midpoint D. In other words, distance AP is two thirds of the distance AD. AP=2/3AD This equality is very similar to the one that Luke said. However, for the first equality to be right, the segments lengths should be swapped. Therefore, Kareem is correct.