EC-1 MATH 310 GEOMETRY: EUCLIDEAN CONSTRUCTIONS WdeS9 Don't just follow the "instructions" . You may see how to make the follow ing constructions on your ow n, and themethods you devise may differ from those given. Give yourself a chance to discover methods yourself. Then readthe “ instructions” . C1. COPY A SEGMENT. Given line segment AB,construct a congruent copy at P.1. Place compass point at A; open com pass t o just span A B. P @2. Place compass point at P (w hich w e can thinkof as A’; draw a circle w ith center P (aka A ’ ). B(Important: don’t change compass opening! This is one of the functions of the compass – it can measure!)A3. Select any point on the circle. Its distance from A’ (= P) is the same as the length ofsegment A B. Use the straightedge to drawthe segment A ’ B’ .CC2. COPY A TRIANGLE. Given a triangle ABC, construct a congruent triangle A'B'C' w ith A'B' on l . 1. Place compass point at A; m easure A B. A B(T : Draw arc through B).2. Mark A’ on l Place compass point at A' ;draw arc through l ; m ark the int ersect ion point B' .3. Now locate C' as follow s:use compass to measure the span of AC 3a. Draw an arc of radius AC, centered at A'3b. Use a similar procedure to draw an arc l of radius BC centered at B'. 3c. The point w here these arcs intersect is C' ...just the right distance from A' & B'. 4. Draw the triangle, connecting A’B’C’.If the sides of one triangle are congruent to the respective sides of a second triangle, the entire triangles are congruent. ("SSS") D E1 . Copy triangle DEF to line lso that the longest side of t he new t riangle, D’ E’ F’ ,lies on line l F El* co n * R9EC-2 M ATH 310 GEOM ETRY: EUCLIDEAN CONSTRUCTIONS Given a line segment AB, construct an equilateral triangle with side AB.E2 .BHow do I do that? What do I know? ...The three sides of the triangle must be congruent.One side is already specified.1. Place compass point at B; draw arc through A.2. Place compass point at A; draw arc t hrough B.3. The point w here these arcs intersect is just the * right distance* from each. Call it point C, the third vert ex of the triangle.4. Draw the triangle! A 1 2 3 E3 . Construct a triangle w hose sides are the lengths of the three segments at right:The theorem that guarantees all such triangles w ill be congruent is the .Can this be done w ith ANY three lengths? (Try 2" & 2" and 5".)C3. COPY AN ANGLE CGiven p ABC, construct a CONGRUENT ANGLEwith vertex X, through Y.1. Place compass point at B; draw an arc w ithdecent radius (NO itsy bitsies!),, intersecting AB at D and BC at E. (Mark D & E.)2. Place compass point at X, and draw an B Aarc matching the one in step 1. Mark D' .(D’ corresponds to D.)3. Placing the compass point at D, draw an arc through E; copy onto the new construction. 4. You have located E'. Draw the angle.(Essentially you created a triangle including angle ABC, then copied it!)X YE4 . Copy the angle at A (pCA B) t o A ’ . Then copy the angle at B (pCBA) to B’ (USE the GIVEN B’ !). Extend the rays so they intersect. Mark the point w here the rays intersect as C’. C ABA’ B’What can you say about the angles of the new triangle, A’B’C’ ?Is triangle A’B’C’ congruent to triangle ABC?* co n * w dR9EC-3 M ATH 310 GEOM ETRY: EUCLIDEAN CONSTRUCTIONS CC4. BISECT an ANGLE Given angle pABC, constructtw o angles with half the measure of pABC.1. Place compass point at B; draw an arc offair radius (NO squinchies!!!), intersecting AB at D and BC at E, as in C3.2. Place compass point at D, and draw anarc through E tow ard AB. 3. Repeat w ith compass point at E, draw ing arc through D tow ard BC. B A4. The point w here these arcs cross is onthe bisecting line, as is B. Draw the line.Why did this w ork? What tw o congruenttriangles w ere created?C5. PERPENDICULAR BISECTORGiven a line segment AB, construct a perpendicular bisector of AB.1. With compass point on A, draw arc throughB, extending arc more than 60 both w ays. A Bo2. Repeat w ith compass point at B, draw ingan arc through A.3. The tw o points w here these arcs crossare on the desired line...because the points on the zbisector are equidistant from A & B. (W hy?)E5 . Given a line l, construct a lineperpendicular to l .(This is just like C5, but less specific. Take advantage of the l construction in C5! i.e. create a segment ! )C6. PERPENDICULAR FROM A POINTGiven a point P and a line l not containing P.the point, construct a perpendicular to the line,through the point.Hint: Compare this task to E5– w here you made your ow n segment. If w e find tw o points on l that are equidistantfrom P, w e'll have a situation similar to C5. lj (There is a very simple alternative method.) * con * w dR9EC-4 M ATH 310 GEOM ETRY: MORE EUCLIDEAN CONSTRUCTIONS C6a. PARALLEL LINEGiven a line "l" and a point P not on l, construct a line through P parallel to l.(Note there are many w ays to accomplish this!)E.g. construct a line k z to l through P; P.then construct a z to k at P. (the hard way)Or: Draw a line t through P intersecting lat point O ... then copy the p, formed by lt and l at O, to P. t P O lOr... Find another w ay to do this .(There is a very simple w ay!)E6 . Construct a parallelogram w ith given three points as vertices.(W hat do you know about parallelograms?)(There are multiple possible outcomes here.) A C B C C CTERM S YOU MUST KNOW (Taking a break from construction here):Before w e move on to the constructions on the next page, let’s make sure w e know the terminology.We often speak of the “ base” of a triangle, view ing it as the side the triangle is ²C“sitting on” , in this case, AB. In fact, any side of the triangle can be the “base” . Identif y t he ot her t w o bases of t his t riangle: A BAn ALTITUDE of a triangle is a segment from the vertex of a triangle, perpendicular to theopposite side, or “base” . For each triangle, there are three altitudes. ²CDraw the missing altitudes. Another exercise: Draw a triangle on a sheet of thin notebook paper. A BFOLD the paper to locate an altitude of the triangle. A M EDIAN of a triangle