Parallel, Perpendicular, Line Bisector and Angle Bisector
Drawing a parallel line
- Select the POINT tool (Window 2) and draw a point O on line r.
- Select the COMPASS tool (Window 6). Then click on O and A (it represents the length opening of
the compass) and again on O (represent the sharp end of the compass).
-Select the INTERSECT tool (Window 3) and mark points B and C, which are intersection points between the circle with the line r.
- Select the COMPASS tool (Window 6). Then click on point B and point A (it represents the length opening of the compass) and again on point C (it represents the sharp end of the compass).
-Select the INTERSECT tool (Window 3) and mark point D, which is the point of intersection of the two circles. -Select the LINE tool (Window 3) and click on A and D. Label this line s.
- Select the SHOW/HIDE OBJECT tool (Window 7) and hide the circles, points B, C and D, leaving only
the lines and point A.
-Select the RELATION (Window 8) and click on the two lines. What happens when you follow all the steps above?
- Select the MOVE tool (Window 1). Move either point A or line r. What can you see?
Parallel Line
Perpendicular line (point not belonging to line)
- Select the COMPASS tool (Window 5). Then click on the segment AB (it represents the length opening of the compass) and on E (represents the sharp end of the compass).
- Select the INTERSECT tool (Window 3) and mark F and G. They are points of intersection of the circle with the line.
- Select the COMPASS tool (Window 6). Then click on point F and point G (it represents the length opening of the compass) and again on point F (it represents the sharp end of the compass). After that, click on point G and point F (it represents the length opening of the compass) and again on G (it represents the sharp end of the compass).
- Select the INTERSECT tool (Window 3) and mark a point H, point of intersection of the last two circles.
-Select the LINE tool (Window 4) and click on point E and point H. The intended perpendicular line will appear. Let us analyse it.
- Select the INTERSECT tool (Window 3) and mark point I, point of intersection of points h and g.
- Select the ANGLE tool (Window 9). Click on points E, I and C to mark the angle EIC (the vertex of the angle will always be the second point clicked). What is the measurement of this angle?
- Select the SHOW / HIDE OBJECT tool (Window 7) and hide the circles, points H, F and G, leaving only the lines and point E.
-Select the RELATION tool (Window 8) and click on the two lines. What happens when you follow all the steps above?
- Select the MOVE tool (Window 1). Move either point E or line g. What can you see?
Perpendicular line (point not belonging to the line)
Perpendicular line (point belonging to line)
-Select the COMPASS tool (Window 5). Then click on the segment AB (it represents the length opening of the compass) and on E (it represents the sharp end of the compass).
- Select the INTERSECT tool (Window 3) and mark the intersections F and G of the circle with the line g.
- Select the COMPASS tool (Window 6). Then click on point F and point G (it represents the length opening of the compass) and again on point F (it represents the sharp end of the compass). After that, click on point G and point F (it represents the length opening of the compass) and again on G (represents the sharp end of the compass).
- Select the INTERSECT tool (Window 3) and mark a point H, point of intersection of the last two circles.
-Select the LINE tool (Window 4) and click on point E and point H. The intended perpendicular line will appear.
-Select the ANGLE tool (Window 9). Click on points H, E and C to mark the angle HEC (the vertex of the angle will always be the second point clicked). What is the measurement of this angle?
- Select the SHOW / HIDE OBJECT tool (Window 7) and hide the circles, points H, F and G, leaving only the lines and point E.
-Select the RELATION tool (Window 8) and click on the two lines. What happens when you follow all the steps above?
- Select the MOVE tool (Window 1). Move either point E or line g. What can you see?
Perpendicular line (point belonging to line)
Line Segment Bisector
- Select the COMPASS tool (Window 6). Then click on point A and point B (it represents the length opening of the compass) and again on point A (it represents the sharp end of the compass). After that click on point B and point A (it represents the length opening of the compass) and again on B (it represents the sharp end of the compass).
- Select the option INTERSECT (Window 3) and mark C and F, which are the intersections between the two circles.
-Select the LINE tool (Window 4) and click on C and D. This is the aimed line bisector.
- Select the INTERSECT (Window 3) and mark E, which is the intersection of g with segment AB.
- Select the SHOW/HIDE OBJECT tool (Window 7) and hide the circles, points C and D, leaving only the lines and point E.
- Select the MOVE tool (Window 1) Move point A or B. What can you see?
Drawing the Line segment Bisector
Analysis 1
Proof that any point of the line has the same distance to A and to B.
Analysis 2
Proof that the angle AEC measures 90º.
Angle Bisector with visible vertex
-Select the COMPASS tool (Window 5). Then click on the segment AB (it represents the length opening of the compass) and on C (it represents the sharp end of the compass).
- Select the INTERSECT tool (Window 3) and mark F and G. They are the points of intersection between the circle and the rays (Half-lines) that form the angle.
- Select the COMPASS tool (Window 6). Then click on the segment AB (it represents the length opening of the compass) and on G (it represents the sharp end of the compass). Then click on segment AB (it represents the length opening of the compass) and on F (It represents the sharp end of the compass).
- Select the INTERSECT tool (Window 3) and mark a point H, point of intersection of the last two circles.
-Select the RAY (HALF-LINE) tool (Window 6) and click on C and H. This is the intended angle bisector.
- Select the SHOW/HIDE OBJECT tool (Window 7) and hide the circles and the points F and G.
-Select the ANGLE tool (Window 7). Click on points D, C and H to mark the angle HEC (the vertex of the angle will always be the second point clicked). Also measure the HCE angle. What can you see?
- Select the MOVE tool (Window 1). Move either point E or line d. What can you see?
Angle Bisector with visible vertex
Angle Bisector (invisible vertex)
In this construction suppose that we want to find the bisector of an angle whose vertex we are not seeing.
-Select the LINE tool (Window 4) and draw a line f so that it intersects the rays (half-lines) i and j.
- Select the INTERSECTION tool (Window 3) and mark H and I, which are the points of intersection of line f with rays (half-lines)i and j, respectively.
-Select the POINT tool (Window 2) and draw a point J on ray i (which is positioned to the left of the line f). Also draw a point K on ray j (which is positioned to the left of line f).
- Select the ANGLE BISECTOR tool (Window 5). Click on J, H and I to create the angle bisector JHI. Also click on K, I and H to draw the angle bisector KIH.
- Select the INTERSECTION tool (Window 3) and mark L, which is the intersection of the two angle bisectors.
- Select the ANGLE BISECTOR tool (Window 5). Click on D, H and E to draw the angle bisector of angle DHE. Also click on E, I and D to draw the angle bisector of angle EID.
- Select the INTERSECT tool (Window 3) and mark L, which is the intersection of the two angle bisectors.
-Select the LINE (Window 4) and click on L and M. This is the desired angle bisector. In order to verify this,
check the HIDE / SHOW VERTEX box.
-Select the MOVE tool (Window 1) Move either point E or line d. What can you see?