THALES' THEOREM

SIMILAR SHAPES

HOW TO DIVIDE A LINE SEGMENTS INTO A NUMBER OF EQUAL PARTS

Measuring with shadows

Using shadows is a quick way to estimate the heights of trees, flagpoles, buildings, and other tall objects. To begin, pick an object whose height may be impractical to measure, and then measure the length of the shadow your object casts. Also measure the shadow cast at the same time of day using a yardstick (or some other object of known height) standing straight up on the ground. Since you know the lengths of the two shadows and the length of the yardstick, you can use the fact that the sun’s rays are approximately parallel to set up a proportion with similar triangles

 

Measuring with a mirror The teacher places the mirror at point C, a distance ds away from the student (see picture). She then steps away from the mirror until she sees the top of the student’s head in the mirror. Let’s call the distance from the teacher to the mirror dt. The teacher knows her height, and she knows that the angle of incidence equals the angle of reflection when a beam of light hits a reflective surface. Since the triangles ABC and DEC are right triangles, and since they share the angle ß, they are similar. So by knowing her own height, and by measuring her own as well as the student’s distance from the mirror, she can calculate the student’s height.