AMSCO'S Geometry. New York - Rye High School

PL h b. Choose point N on . Construct Q on PL hsuch that PQ 8PN. Locate S on PQ suchThe Meaning of Locus 60911. a. Draw an angle, LPR.that PS : SQ 3 : 5.c. Draw QR and construct a line through S parallel to QR and intersecting PR hat T.3d. Prove that PST PQR with a constant of proportionality of 8.14-2 THE MEANING OF LOCUSIn a construction, the opening between thepencil and the point of the compass is a fixeddistance, the length of the radius of a circle.The point of the compass determines a fixedpoint, point O in the diagram. If the length ofthe radius remains unchanged, all of thepoints in the plane that can be drawn by thecompass form a circle, and any points thatcannot be drawn by the compass do not lie onthe circle. Thus, the circle is the set of allpoints at a fixed distance from a fixed point.This set is called a locus.ODEFINITIONA locus is the set of all points that satisfy a given condition or set of conditions.The example of the circle given above helps us to understand what the definitionmeans. Every definition can be written as a biconditional:p: A point is on the locus.q: A point satisfies the given conditions.1. (p → q): If a point is on the locus, then the point satisfies the given conditions.All points on the circle are at a given fixed distance from the center.2. (p → q): If a point is not on the locus, then the point does not satisfythe given conditions. Any point that is not on the circle is not at the givendistance from the center.Recall that the statement (p → q) is the inverse of the statement(p → q). A locus is correct when both statements are true: the conditional andits inverse.We can restate the definition of locus in biconditional form: A point P is a point of the locus if and only if P satisfies the given conditionsof the locus.