can we draw a perpendicular line without using a compass ?
The procedure to construct a perpendicular to a line using a ruler and set-square is given in lesson 3 of the same chapter. The explanation of the construction has been provided in the study material embedded with animation for your understanding. Go through the study material and write if you encounter any difficulty in understanding the construction.
This page shows how to construct a perpendicular to a line through an external point, using only a compass and straightedge or ruler. It works by creating a line segment on the given line, then bisecting it. The bisector will be a right angles to the given line. (See proof below).
|Step-by-step Instructions||Printer friendly version|
|After doing this||Your work should look like this|
|Start with a line and point R which is not on that line.|
|Step 1||Place the compass on the given external point R.|
|Step 2||Set the compass width to a approximately 50% more than the distance to the line. The exact width does not matter.|
|Step 3||Draw an arc across the line on each side of R, making sure not to adjust the compass width in between. Label these points P and Q|
|Step 4||At this point, you can adjust the compass width. Recommended: leave it as is. |
From each point P,Q, draw an arc below the line so that the arcs cross.
|Step 5||Place a straightedge between R and the point where the arcs intersect. Draw the perpendicular line from R to the line, or beyond if you wish.|
|Step 6||Done. This line is perpendicular to the first line and passes through the point R. It also bisects the segment PQ (divides it into two equal parts)|
The image below is the final drawing above with the red lines added.
|1||Segment RP is congruent to RQ||They were both drawn with the same compass width|
|2||Segment SQ is congruent SP||They were both drawn with the same compass width|
|3||Triangle RQS is congruent to triangle RPS||Three sides congruent (SSS), RS is common to both.|
|4||Angle JRQ is congruent to JRP||CPCTC. Corresponding parts of congruent triangles are congruent.|
|5||Triangle RJQ is congruent to triangle RJP||Two sides and included angle congruent (SAS), RJ is common to both.|
|6||Angle RJP and RJQ are congruent||CPCTC. Corresponding parts of congruent triangles are congruent.|
|7||Angle RJP and RJQ are 90°||They are congruent and supplementary (add to 180°).|