Locus (1963)

6358. Take one. Pelican. [Music] [Applause] An illegal radio is transmitting signals from somewhere in this area. From the strength of the signals received at the detector site, the distance between that point and the source is estimated at 3/4 of a mile. But in which direction? How is the exact location to be pinpointed? First, all of the possible locations on that 3/4 mile circle are plotted. Additional information received shows that the radio is one block west of the river. Again, where? Plotting all of these possible positions gives two points of intersection with a circle A and B. Since A is a vacant lot, B is raided and the operation of the radio is halted. [Music] The set of all positions that a point may occupy when it is subject to one or more conditions is called a locust. [Music] In the problem just solved, one locus had the shape of a circle, the other locus a straight line and their intersection two points. These loi are the basic elements of all the shapes we encounter in high school geometry. Though we limit our study of locust to those sets of points which are circles, straight lines or individual points, we will see that other shapes are possible. [Applause] To illustrate the first of five basic locus theorems, look at this football cheering section holding cards. Light on one side, dark on the other. On signals from the leader, certain cards are flipped up to form allover patterns, including instructions to the team, school mascot portraits, instructions to the opposing team. Suppose the instructions were for each card 20 ft from the center to flip. The set of cards constitutes the locus of cards that are 20 ft from the center and its shape is a circle with a radius of 20 ft. The knob of an opening door has a set of positions in the shape of an arc of a circle. The locus of this knob is an arc of a circle. Due to the Earth's rotation on its axis, stars in the sky appear to rotate around the North Star. The locus of any star over a period of 24 hours would have the shape of a complete circle. expressed in the language of geometry. The first basic locus theorem is stated. The locus of points a given distance r from a point p consists of the set of points on a circle with center p and radius r. We can find the locus of the vertex of a triangle with a fixed base in which the median to that base is of constant length. The locus has the shape of a circle. Just as the locus of a point in a triangle may have the shape of a circle, so may a point in a circle have a locus in the shape of a straight line. Under certain conditions, the locus of this wheel's hubcap is a straight line. Using this line as a fence to separate our cheering section, we illustrate the second basic locus theorem. Instructions are now given for all cards, which are four roads on either side of the fence, to flip. If the fence were a vertical division, the locust would result in these parallel lines. The same set of positions could be the locus of a treasure buried 5 ft from a given fence or the locus of a car 3 ft from the center line. In geometric terms, we state our second basic locus theorem. The locus of points a given distance D from a given line L consists of the set of points on two lines parallel to L and distance D on each side of it. As an application, consider a triangle with a fixed base and a constant altitude. We find the locus of the vertex consists of a set of points on the two lines parallel to the base. Our cheerleader now gives instructions to all cards which are the same distance from A and B to flip. The resulting set of points is the perpendicular bis sector of line A B. A piece of paper marked A and B if folded so that A coincides with B will show that line O A equals line O B PA equals PB and QA equals QB. When opened we see the locus of points equidistant from A and B consists of the set of points on crease XY. the locus of footprints equidistant from two trees or a ship passing equidistant from two boys. And in the language of geometry, we state the third basic locus theorem. The locus of points equidistant from two given points A and B consists of the set of points on the perpendicular bis sector of segment AB. With segment AB as the fixed base of a set isles triangle, we can find the locus of the vertex. The locus consists of a set of all points on the perpendicular bis sector of segment AB. Instructions to our cheering section are every card as far from the left rail as from the right flip. The locus of cards equidistant from the two parallel rails consists of the set of cards parallel to each rail and midway between them. A maintenance truck painting a white line moving equidistant from the sides of a road or a plane taxiing equidistant from the sides of a lighted runway. Each indicates a set of positions which when stated in the language of geometry becomes our fourth basic locus theorem. The locus of points equidistant from two parallel lines consists of the set of points on the line parallel to them and midway between them. As an application for a circle which is tangent to each of two parallel lines, we can see that the locus of the center is this line. To illustrate our final basic locus theorem, our cheering section is told to flip all cards which are as far from the front rail as from the center rail. The resulting locus consists of a set of cards that bisect angles formed by the rails. You can construct the locus of points equidistant from EA and EF by folding this paper so that they coincide. Unfolded, P is as far from EA as from EF and angle one will equal angle two. The crease EB is the locus of points equidistant from EA and EF. It bicts angle AEF. By repeating this process, we obtain the complete locus of points equidistant from EA and EF. The given lines need not be perpendicular to each other. Expressed in the language of geometry, the fifth basic locus theorem is stated, the locus of points equidistant from two intersecting lines consists of the bis sectors of the angles formed by the lines. As an application, the locus of points within a triangle equidistant from sides AB and AC consists of the set of points on the bis sector of angle B A. Some locus applications involve satisfying two conditions simultaneously. For example, we could have stated our problem of the illegal transmitter in this way. Find the locus of points 3/4 of a mile from point P and one block west of the river. The locus consists of the two points common to the circle and the line. In general, to find the locus of points that satisfy two conditions, consider one condition at a time and determine the intersection of the resulting lossi. As an example, to construct a circle through the vertices of triangle A B C, find O so that O A equals O B equals O C. In the language of geometry, find the locus of points equidistant from A and B and also equidistant from A and C. The first condition is satisfied by the set of points on the perpendicular bis sector of segment AB. The second by the set of points on the perpendicular bis sector of segment AC. Both conditions are satisfied by the intersection of the two sets at O. Circles, lines or points are not the only shapes which LOSI can take. Science and nature reveal a great variety of other shapes. For example, the parabola, the ellipse, the hyperola, the cyclloid. Planning a moonshot requires the solution of a locust problem with at least two conditions, the locus of the moon and the locust of the rocket. The study of locus deals with a problem of determining the set of all positions that a point may occupy when it is subject to one or more conditions. Let's review our five basic locus theorems. The locus of points a given distance r from a point P consists of the set of points on a circle with center P and radius R. The locus of points a given distance D from a given line L consists of the set of points on two lines parallel to L and distance D on each side of it. The locus of points equidistant from two given points A and B consists of the set of points on the perpendicular bis sector of segment AB. The locus of points equidistant from two parallel lines consists of the set of points on the line parallel to them and midway between them. And finally, the locus of points equidistant from two intersecting lines consists of the bis sectors of the angles formed by the lines. [Music]