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Basic Geometric Constructions When constructing basic geometric figures, you cannot use a ruler to find the lengths of the sides and estimate the angles. Basic geometric constructions require the use of a geometric compass. One type of compass has a point on one side and a pencil on the other side (see figure below). Figure of Geometric Compass The following is a list of basic geometric constructions that we cover. Please watch the video to see how these constructions are done. • • • • • • • Congruent segments Congruent angles Perpendicular bisector Angle bisector Constructing Congruent Segments Have the same length Construction based on congruent radii of congruent circles Constructed using following steps • Put point of compass on one end and pencil tip on the other end and draw an arc • Draw a new point somewhere • Place the point of compass on new point and draw arc of equal length • Draw segment from point to arc Original Line Segment Copied Line Segment © LaurusSoft, Inc. 2010 • • Constructing Congruent Angles Have the same measure between sides Constructed using following steps • Copy one of the segments using process above • Pick a point on the copied segment and draw an arc through that point • Using the intersection of the arc you just drew and the chosen point, create a second arc that intersects the first arc • Copy the other segment that makes up the angle Original Angle Copied Angle B’ B 5 5 C’ C 4 4 A A’ Steps for Constructing Congruent Triangles 1. follow the steps above 2. Draw in third side • • Perpendicular Bisector of a Line Segment Forms 90o angle Cuts through midpoint of the segment This construction can be used whenever you are needing to locate the midpoint or needing to construct a line perpendicular to another line. © LaurusSoft, Inc. 2010 • • • • • • • • • Steps for Constructing Perpendicular Bisector Put point of compass at one of the endpoints Expand compass more than half way and draw large arc Put point of compass at other endpoint Keeping the compass at the same distance, draw another large arc (the arcs should intersect each other above and below the segment) Draw a line segment through the two points of intersection of the two arcs Constructing Angle Bisector Put point of compass at vertex and draw an arc that cuts through both sides of angle Put point of compass at one intersection point and draw arc in middle of angle Keeping the compass at the same distance, draw another arc from the other point of intersection (the arcs should intersect each other inside the angle) Draw a line segment through the vertex and intersection of the two arcs inside the angle B C A © LaurusSoft, Inc. 2010 Points of Concurrency Points of concurrency are the points where certain line segments constructed from the parts of a triangle intersect. You should be able to locate all of these points through constructions. Incenter – intersection of the three angle bisectors (construct all three angle bisectors) Circumcenter – intersection of the three perpendicular bisectors (construct all three perpendicular bisectors) Centroid – intersection of the three medians (construct all three perpendicular bisectors to find the midpoints of each side and connect each midpoint to opposite vertex) Orthocenter – intersection of the three altitudes (see video for this construction) © LaurusSoft, Inc. 2010