2 Don’t change your radius! Construction #1Construct a segment congruent to a given segment.This is our compass.Given:ABProcedure:1. Use a straightedge to draw a line. Call it l.Construct: XY = ABDon’t change your radius!2. Choose any point on l and label it X.3. Set your compass for radius AB and make a mark on the line where B lies. Then, move your compass to line l and set your pointer on X. Make a mark on the line and label it Y.lXY
3 Construct an angle congruent to a given angle Construction #2Construct an angle congruent to a given angleACBGiven:Procedure:D1) Draw a ray. Label it RY.2) Using B as center and any radius, draw an arc intersecting BA and BC. Label the points of intersection D and E.EConstruct:3) Using R as center and the SAME RADIUS as in Step 2, draw an arc intersecting RY. Label point E2 the point where the arc intersects RYD2RY4) Measure the arc from D to E.E25) Move the pointer to E2 and make an arc that that intersects the blue arc to get point D26) Draw a ray from R through D2
4 Bisector of a given angle? Construction #3How do I construct aBisector of a given angle?CABZGiven:XYProcedure:Using B as center and any radius, drawand arc that intersects BA at X and BC at point Y.2. Using X as center and a suitable radius, draw and arc.Using Y as center and the same radius, draw an arcthat intersects the arc with center X at point Z.3. Draw BZ.
5 How do I construct a perpendicular bisector to a given segment? Construction #4How do I construct a perpendicularbisector to a given segment?XGiven:ABYProcedure:Using any radius greater than 1/2 AB, draw four arcs ofequal radii, two with center A and two with center B.Label the points of intersection X and Y.Draw XY
6 Construction #5 How do I construct a perpendicular bisector to a given segment at a given point?ZGiven:CkXYProcedure:Using C as center and any radius, draw arcs intersectingk at X and Y.Using X as center and any radius greater than CX, draw an arc. Using Y as center and the same radius,draw and arc intersecting the arc with center X at Z.Draw CZ.
7 Construction #6 How do I construct a perpendicular bisector to a given segment at a given point outside the line?kPGiven:XYZProcedure:Using P as center, draw two arcs of equal radii thatintersect k at points X and Y.Using X and Y as centers and a suitable radius, draw arcsthat intersect at a point Z.Draw PZ.
8 Construction #7 How do I construct a line parallel to a given line through a given point?kP1lGiven:ABProcedure:Let A and B be two points on line k. Draw PA.At P, construct <1 so that <1 and <PAB are congruentcorresponding angles. Let l be the line containing theray you just constructed.
9 Concurrent LinesIf the lines are Concurrent then they all intersect at the same point.The point of intersection is called the “point of concurrency”
10 What are the 4 different types of concurrent lines for a triangle? Perpendicular bisectorsAngle bisectorsAltitudesMediansCircumcenterIncenterOrthocenterCentriodConcurrent LinesPoint of ConcurrencyCircumcenterSPOrthocenterSPIncenterSPCentroidSP
11 Given a point on a circle, construct the tangent to the circle at the given point . Construction #8PROCEDURE:Given: Point A on circle O.1) Draw Ray OA2) Construct a perpendicular through OA at point A.53Now, using the same radius, construct arcs 5 & 6 using point P as the center so that they intersect arcs 3 & 4 to get points X & YConstruct arcs 3 & 4 using point Q as the center and any suitable radius (keep this radius)X3) Draw tangent line XY21OAQPConstruct arcs 1 and 2 using any suitable radius and A as the centerY64
12 Construction #9Given a point outside a circle, construct a tangent to the circle from the given point.PROCEDURE:Given: point A not on circle O1) Draw OA.X312) Find the midpoint M of OA (perpendicular bisector of OA)3) Construct a 2nd circle with center M and radius MAMOA4) So you get points of tangency at X & Y where the arcs intersect the red circleConstruct arcs 1& 2 using a suitable radius greater than ½AO( keep this radius for the next step)Construct arcs 3& 4 using the same radius(greater than ½AO)You get arcs 5 & 65) Draw tangents AX & AYY42
13 Construction #10 Given a triangle construct the circumscribed circle. PROCEDURE:Given: Triangle ABC78ABC561) Construct the perpendicular bisectors of the sides of the triangle and label the point of intersection F.3412Bisect segment BC; Using a radius greater than 1/2BC from point C construct arcs 5 & 6From point B construct arcs 7 & 8 and draw a line connecting the intersections of the arcs2) Set your compass pointer to point F and the radius to measure FC.F3) Draw the circle with center F , that passes through the vertices A, B, & CradiusNow construct the perpendicular bisector of segment AB and label point F, where the 3 lines meet.Bisect segment AC; Using a radius greater than 1/2AC from point C construct arcs 1 & 2From point A construct arcs 3 & 4 and draw a line connecting the intersections of the arcs
14 Construction #11 Given a triangle construct the inscribed circle. PROCEDURE:Given: Triangle ABCABC1) Construct the angle bisectors of angles A, B, & C, to get a point of intersection and call it F2) Construct a perpendicular to side AC from point F, and label this point G.F3) Put your pointer on point F and set your radius to FG.4) Draw the circle using F as the center and it should be tangent to all the sides of the triangle.GXY
15 Three congruent segments Remember you made 3 because you are dividing by 3, but if you wanted to divide by, say, 6 you would have to make 6 congruent parts on the ray and so on for 7,8,9…Construction #12Given a segment, divide the segment into a given number of congruent parts.Given: Segment ABPROCEDURE:Divide AB into 3 congruent parts.1) Construct ANY RAY from point A that’s not ABABDC2) Construct 3 congruent segments on the ray using ANY RADIUS starting from point A. Label the new points X, Y, & ZX3) Draw segment ZB and copy the angle AZB ( 1) to vertices X & YSo AC=CD=DBThree congruent segmentsY1Use any suitable radius that will give some distance between the points4) Draw the the rays from X & Y, they should be parallel to the segment ZB and divide AB into 3 congruent parts.ZRemember keep the same radius!!
16 Construction #13Given three segments construct a fourth segment (x) so that the four segments are in proportion.abcGiven:Construct: segment x such thatacxbbxPROCEDURE:a1) Using your straight edge construct an acute angle of any measure.1N2) On the lower ray construct “a” and then “c “ from the end of “a”.c3) On the upper ray construct “b” and then connect the ends of “a & b”4) Next copy angle 1 at the end of “c” and then construct the parallel line
17 Construction #14 X Given 2 segments construct their geometric mean. Make sure to set your radius to morethan 1/2NM then:Mark off 2 arcs from M. Keep the sameradius and mark off 2 more arcs from N,crossing the first two.Draw the perpendicular bisector PQthrough point OMark off 2 arcs from Y.Keep the same radius and mark off 2 more arcs from X, crossing the first two.abGiven:Construct: segment x such that:Mark off 2 equal distances on eitherside of point O using any radius andthen bisect this new segment(I used the distance from O to M,but remember any radius will do)axbThe orange segment is x the geometric mean between the lengths of a and bPROCEDURE:9731XQK1) Draw a ray and mark off a+b.2) Bisect a+b (XY) and label point M.65XYM3) Construct the circle with center M and radius = MY (or MX)aNOb4) Construct a perpendicular where segment “a” meets segment “b” (point O)81024PL
18 The Meaning of LocusIf a figure is a locus then it is the set of all points that satisfy one or more conditions.The term “locus” is just a technical term meaning “a set of points”.So , a circle is a locus.Why??Because it is a set of points a given distance from a given point.
19 What can a locus be??Remember it is a SET OF POINTS so if you recall the idea of sets from algebra it is possible for a set to be empty.So a set could be:The empty set. (no points fit the condition or conditions)A single point.Two points, three points….An infinite set of points. (like a line, circle, curve,…)