Geometry Unit 6: Constructions

Midpoint & Perpendicular Bisector

1) Place the compass point at A. Use a compass setting greater than half the length of AB. Draw an arc. 2) Keep the same compass setting. Place the compass point at B. Draw an arc. It should intersect the other arc in two places. 3) Use a straightedge to draw a segment through the points of intersection. This segment bisects AB at M , the midpoint of AB. This segment is also perpendicular to AB.

Angle Bisector

1) Place the compass point at C. Draw an arc that intersects both sides of the angle. Label the intersections A and B. 2)Place the compass at A. Then place the compass point at B. Using the same compass setting, draw another arc. 3) Label the intersection D. Use a straightedge to draw a ray through C and D. This is the angle bisector.

Perpendicular to a line through a point not on the line

1) Place the point of your compass on the point and use it as the center of an arc that intersects the line at two points. Your original point will lie on the perpendicular bisector of the segment formed by these two new points. 2) Construct a perpendicular bisector between the arc on the line.

Inscribed Equilateral Triangle

1) Complete the same steps for constructing a regular hexagon. 2) When connecting the vertices, draw a line between every other vertex, for a total of three lines.

Inscribed Square

1) Draw a diameter. 2) Draw a perpendicular bisector to the diameter. 3) Connect the four points of intersection with the circle-these make 4 vertices.

Copy an Angle

1) Draw a line. Label the point on the line D. 2) Draw an arc with the center A. label B and C. with the same radius. draw an arc with center D. Label E. 3) Draw an arc with radius BC and center E. Label the intersection F. 4) Draw DF. <EDF = <BAC.

Two targets to circle through a point

1) Draw a straight line between the center O of the given circle and the given point P. 2) Find the midpoint of this line by constructing the line's perpendicular bisector. 3) Place the compasses on the midpoint just constructed, and set its width to the center O of the circle. 4) Without changing the width draw an arc across the circle in the 2 possible places. These are the contact points J,K for the tangents. 5) Draw 2 tangent lines from P through J and K.

Parallel Lines

1) Draw points Q and R on m. Draw PQ. 2) Draw an arc with the compass point at Q so that it crosses QP and QR. 3) Copy <PQR on QP as shown. Be sure the two angles are corresponding. Label the new angle <TPS. 4) Draw PS. Because <TPS and <PQR are congruent corresponding angles, PS II QR.

Inscribed Regular Hexagon

1) Mark a point anywhere on a circle. This will be the first vertex of the hexagon. 2) Set the compass on the point and set the width of the compass to the center of the circle O (the radius) 3) Make an arc across the circle the distance of the radius. This will be the next vertex of the hexagon. 4) Move the compass on the next vertex and draw another arc. This will be the third vertex of the hexagon. 5) Continue in this way until you have all 6 vertices. 6) Draw a line between each successive pairs of vertices, for a total of 6 lines.

Perpendicular to a line through a point on the line

1) Place the compass at P. Draw an arc that intersects line m twice. Label the intersections A and B. 2) Use a compass setting greater than AP. Draw an arc from A. With the same setting, draw an arc from B. Label the intersection of the arcs as C. 3) Use a straightedge to draw CP. This line is perpendicular to line m and passes through P.

Copy a Line

1) Use a straightedge to draw a segment longer than AB. Label the point C on the new segment. 2) Set your compass at the length of AB. 3) Place the compass point at C and mark the second point D on the new segment. CD is congruent to AB.