Session 2: Construction of Perpendicular Lines (Perpendicular Bisector) – Steps and Interactives

Constructions are step-by-step processes used to create accurate geometric figures. To create a construction by hand, there are a few tools that you can use: 

1. Compass: A device that allows you to create a circle with a given radius. Not only can compasses help you to create circles, but also they can help you to copy distances.

2. Straightedge: Anything that allows you to produce a straight line. In geometric constructions, a straightedge should not be used to measure distances. An index card works well as a straightedge.

3. Set Squares: Two triangular-shaped instruments primarily used for drawing perpendicular and parallel lines. In one, the angles are 30∘,60∘and 90∘and in the other, the angles are 90∘,45∘and 45∘.

4. Paper: When a geometric figure is on a piece of paper, the paper itself can be folded to construct new lines. 

A few important terms:

• Perpendicular lines are lines that intersect at right angles. 
• To bisect a segment or an angle means to divide it into two congruent(equal) parts. 
• A bisector of a line segment will pass through the midpoint of the line segment. 
• A perpendicular bisector of a segment passes through the midpoint of the line segment and is perpendicular to the line segment. 

Construct a Perpendicular on a Line at a Given Point using Ruler and Set Square

To construct a perpendicular on a line at a given point using a ruler and set square, we follow the following steps:

Step 1: Create a line using a ruler and a pencil and label it, say AB.

Step 2: Place the ruler horizontally along AB. This ruler should be kept fixed.

Step 3: Place the set square along the edge of the ruler so as to slide easily against the ruler.

Step 4: Starting from point A, slide the set square along the ruler until the right angle vertex coincides with the given point X.

Step 5: Draw a line along the edge of the other arm of the set square at point X. The line XY so obtained is the required line perpendicular to the given line AB at a given point X.

Construct a Perpendicular on a Line and Passing Through a Given Point not Lying on it Using Ruler and Set Square

To construct a perpendicular on a line and passing through a given point not lying on it using a ruler and set square, we follow the following steps:

Step 1: Create a line using a ruler and a pencil and label it, say AB.

Step 2: Place the set square horizontally along line AB.

Step 3:  Place the ruler along the edge opposite to the right angle of the set square so as to slide easily against the set square.

Step 4: Hold the ruler firmly and slide the set square along the ruler until the vertex of the right angle coincides with the given point P.

Step 5: Draw a line along the edge of the other arm of the set square at point P. The line PQ so obtained is the required line perpendicular to the given line AB and passing through a given point P not lying on it.

Construct a Perpendicular on a Line at a Given Point Using Ruler and Compass

To construct a perpendicular on a line at a given point using a ruler and a compass, we follow the following steps:

Step 1: Draw a line and mark a point P lying on it.

Step 2: Start by using your compass to draw a partial circle centred at the point P that intersects the original line at two points, A and B.

Step 3: Now, place the compass at one endpoint of the line segment. Adjust the width of the compass to slightly longer than half the line segment length. With A as the centre, draw an arc. Similarly, with B as the centre and the same compass width, draw another arc to intersect the first. Point Q, where the arcs intersect, is the point of intersection.

Step 4: Use your straightedge to draw a line connecting the intersection point Q and point P. Line PQ is perpendicular to the original line and passes through the given point P .