Perpendicularity

Author:: Jorge Cássio

Topic:: Angles, Constructions, Straight Lines

Perpendicular Lines - Definition

Two lines are perpendicular if, and only if, they are concurrent and form complementary adjacent congruent angles. These lines form right angles (90º).

Question 1

Which pairs of straight lines are perpendicular?

Select all that apply

A
a and b, b and f, a and f.
B
only a and b
C
It is impossible to answer.
D
None of them

Check my answer (3)

Oblique lines - Definition

Two straight lines are oblique when they are concurrent, but not perpendicular.

Move point A

Constructing a Perpendicular line from a point outside the line.

In the following GeoGebra applet, follow the steps below: - Select the COMPASS (Window 5). Then click on the segment AB (opening of the compass) and on E (compass point). - Select the option INTERSECT (Window 3) and mark the intersections F and G of the circumference with the line g. - Select the COMPASS (Window 6). Then click on point F and point G (it will open the compass) and again on point F (it will close the compass and form a circle). After that, click on point G and point F (it will open the compass) and again on G (it will close the compass and form a second circle). - Select the option INTERSECT (Window 3) and mark a point H, point of intersection of the last two circunferences. -Select the option LINE (Window 4) and click on point E and point H. It will create the intended perpendicular line. Let us analyse it. - Select the option INTERSECT (Window 3) and mark point I, point of intersection of points h and g. - Select the option ANGLE (Window 6). 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 option SHOW / HIDE OBJECT (Window 7) and hide the circles, points H, F and G, leaving only the lines and point E. -Select the option RELATION tool (Window 8) and click on the two lines. What happens? - Select the option MOVE (Window 1) move point E or line g. What can you see?

Analysis 1

Write an argument to justify the construction. Use the perpendicular bisector property:

line that passes perpendicularly through the midpoint;
geometric location of points equidistant (same length) from the endpoints of a segment.

Construction of the Perpendicular from a point on a line

- Select the COMPASS (Window 5). Then click on the segment AB (opening of the compass) and on E (compass point). - Select the option INTERSECT (Window 3) and mark the intersections F and G of the circumference with the line g. - Select the COMPASS (Window 6). Then click on point F and point G (it will open the compass) and again on point F (it will close the compass and form a circle). After that, click on point G and point F (it will open the compass) and again on G (it will close the compass). - Select the option INTERSECT (Window 3) and mark a point H, point of intersection of the last two circunferences. -Select the option LINE (Window 4) and click on point E and point H. It will create the intended perpendicular line. - Select the option ANGLE (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 option SHOW / HIDE OBJECT (Window 7) and hide the circles, points H, F and G, leaving only the lines and point E. -Select the option RELATION tool (Window 8) and click on the two lines. What happens? - Select the option MOVE tool (Window 1) and move point E or line g. What can you see?

Analysis 2

Write an argument to justify the construction. Repeat the use of Perpendicular bisector properties.