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Chapter 10

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Construct a segment congruent to a given segment. Given: AB Procedure: 1. Use a straightedge to draw a line. Call it l. 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. Construct: XY = AB XY l Construction #1 This is our compass. Dont change your radius!

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Construct an angle congruent to a given angle Given:Procedure: 1) Draw a ray. Label it RY. Construct: E D 2) Using B as center and any radius, draw an arc intersecting BA and BC. Label the points of intersection D and E. 4) Measure the arc from D to E. E2E2 Construction #2 R Y 3) Using R as center and the SAME RADIUS as in Step 2, draw an arc intersecting RY. Label point E 2 the point where the arc intersects RY 5) Move the pointer to E 2 and make an arc that that intersects the blue arc to get point D 2 D2D2 A C B 6) Draw a ray from R through D 2

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How do I construct a Bisector of a given angle? Given: C A B Procedure: 1.Using B as center and any radius, draw and arc that intersects BA at X and BC at point Y. X Y 2. Using X as center and a suitable radius, draw and arc. Using Y as center and the same radius, draw an arc that intersects the arc with center X at point Z. Z 3. Draw BZ. Construction #3

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How do I construct a perpendicular bisector to a given segment? Given: Procedure: 1.Using any radius greater than 1/2 AB, draw four arcs of equal radii, two with center A and two with center B. Label the points of intersection X and Y. A B X Y 2. Draw XY Construction #4

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How do I construct a perpendicular bisector to a given segment at a given point? Given: C k Procedure: 1.Using C as center and any radius, draw arcs intersecting k at X and Y. XY 2.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. Z 3. Draw CZ. Construction #5

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How do I construct a perpendicular bisector to a given segment at a given point outside the line? Given: k P Procedure: 1.Using P as center, draw two arcs of equal radii that intersect k at points X and Y. XY 2.Using X and Y as centers and a suitable radius, draw arcs that intersect at a point Z. Z 3. Draw PZ. Construction #6

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How do I construct a line parallel to a given line through a given point? Given: k P Procedure: 1.Let A and B be two points on line k. Draw PA. AB 2.At P, construct <1 so that <1 and

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Concurrent Lines If the lines are Concurrent then they all intersect at the same point. The point of intersection is called the point of concurrency

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What are the 4 different types of concurrent lines for a triangle? I.Perpendicular bisectors II.Angle bisectors III.Altitudes IV.Medians Circumcenter Incenter Orthocenter Centriod Concurrent LinesPoint of Concurrency SP Circumcenter Incenter Orthocenter Centroid

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Construction #8 Given a point on a circle, construct the tangent to the circle at the given point. 1) Draw Ray OA 2) Construct a perpendicular through OA at point A. O A Given: Point A on circle O. Construct arcs 1 and 2 using any suitable radius and A as the center 12 P Q Construct arcs 3 & 4 using point Q as the center and any suitable radius (keep this radius) Now, 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 & Y X Y 3) Draw tangent line XY PROCEDURE:

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Construction #9 Given a point outside a circle, construct a tangent to the circle from the given point. O A Given: point A not on circle O PROCEDURE: 1) Draw OA. 2) Find the midpoint M of OA (perpendicular bisector of OA) Construct arcs 1& 2 using a suitable radius greater than ½AO ( keep this radius for the next step) 1 2 Construct arcs 3& 4 using the same radius (greater than ½AO) You get arcs 5 & M 3) Construct a 2 nd circle with center M and radius MA 4) So you get points of tangency at X & Y where the arcs intersect the red circle X Y 5) Draw tangents AX & AY

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Construction #10 Given a triangle construct the circumscribed circle. Given: Triangle ABC PROCEDURE: 1) Construct the perpendicular bisectors of the sides of the triangle and label the point of intersection F. A B C Bisect segment AC; Using a radius greater than 1/2AC from point C construct arcs 1 & 2 From point A construct arcs 3 & 4 and draw a line connecting the intersections of the arcs Bisect segment BC; Using a radius greater than 1/2BC from point C construct arcs 5 & 6 From point B construct arcs 7 & 8 and draw a line connecting the intersections of the arcs F 2) Set your compass pointer to point F and the radius to measure FC. 3) Draw the circle with center F, that passes through the vertices A, B, & C Now construct the perpendicular bisector of segment AB and label point F, where the 3 lines meet. r a d i u s

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Construction #11 Given a triangle construct the inscribed circle. Given: Triangle ABC PROCEDURE: A B C 1) Construct the angle bisectors of angles A, B, & C, to get a point of intersection and call it F 2) Construct a perpendicular to side AC from point F, and label this point G. 3) 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. F G XY

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Construction #12 Given a segment, divide the segment into a given number of congruent parts. Given: Segment AB PROCEDURE: 4) Draw the the rays from X & Y, they should be parallel to the segment ZB and divide AB into 3 congruent parts. Divide AB into 3 congruent parts. 1) Construct ANY RAY from point A thats not AB 2) Construct 3 congruent segments on the ray using ANY RADIUS starting from point A. Label the new points X, Y, & Z 3) Draw segment ZB and copy the angle AZB ( 1) to vertices X & Y 1 Use any suitable radius that will give some distance between the points AB X Y Z Remember keep the same radius!! DC So AC=CD=DB 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…

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Construction #13 Given three segments construct a fourth segment (x) so that the four segments are in proportion. Given: PROCEDURE: 1) Using your straight edge construct an acute angle of any measure. 2) On the lower ray construct a and then c from the end of a. 3) 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 a b c Construct: segment x such that a cx b a c bx 1

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Given 2 segments construct their geometric mean. Construction #14 Given:Construct: segment x such that: a x x b PROCEDURE: 1) Draw a ray and mark off a+b. 3) Construct the circle with center M and radius = MY (or MX) 4) Construct a perpendicular where segment a meets segment b (point O) a b L N P Q Y O X 3 a b K M 2) Bisect a+b (XY) and label point M. The orange segment is x the geometric mean between the lengths of a and b X Mark off 2 arcs from Y. Keep the same radius and mark off 2 more arcs from X, crossing the first two. Mark off 2 equal distances on either side of point O using any radius and then bisect this new segment (I used the distance from O to M, but remember any radius will do) Make sure to set your radius to more than 1/2NM then: Mark off 2 arcs from M. Keep the same radius and mark off 2 more arcs from N, crossing the first two. Draw the perpendicular bisector PQ through point O

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The Meaning of Locus 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. If a figure is a locus then it is the set of all points that satisfy one or more conditions.

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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: A.The empty set. (no points fit the condition or conditions) B.A single point. C.Two points, three points…. D.An infinite set of points. (like a line, circle, curve,…)

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Examples of a Locus in a Plane

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Locus Problems

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