CHAPTER 5

5.1 Exercise Set 

Identify the lines below 

5.2 Construction of Perpendicular Lines. 

Activity : Construction of perpendicular line from an external point to a given line. 

Given line segment AB and point P outside the line, construct a perpendicular line from point P to line AB. 

STEPS 

1. Taking the centre as P and any radius, Place the campus at point P and draw two arcs to cut line AB . 

2. Taking A as the centre and any radius, draw an arc below or above the line opposite point P . 

3. Without changing the radius and taking B as the centre, draw an arc to intersect the previous arc at point Q. 

4. Join the intersection of the arcs from point P to Q.

Figure 5.1: Construction of perpendicular line from an external point to a given line 

Activity : Construction of a Perpendicular line to a given point on a given line segment. 

Given line AB and point P on AB. construct a perpendicular line from point P on AB 

STEPS 

1. Taking P as the centre and any radius, draw two arcs on either side of P name the arcs A and B . 

2. Taking A as the centre and any radius draw an arc either above or below the line. 

3. Without changing the radius and taking B as the centre draw an arc to meet the previous arc at point Q 

4. Join the intersection of the arcs to from P to Q 

Figure 5.2: Construction of a Perpendicular line to a given point on a given line segment 

Activity : Construction of a Perpendicular Bisector. 

Given line segment AB ,draw a perpendicular bisector. 

STEPS 

1. Draw a line segment with end points A and B . 

2. Place the point of compass at A ,stretch out the compass until more than half of the length of AB. 

3. Draw an arc on either side of the line segment. 

4. Keeping the radius of the compass constant, place the point of the compass at point B and draw an arc on either sides of the line segment. 

5. Join the intersection of the arc at point C and D.

Figure 5.3: Construction of a Perpendicular bisector 

5.3  Construction of parallel lines.

STEPS.

1. Draw a line segment with end points AB and a point C outside the line . 

2. Draw an arc at point A taking AB as radius and C as the centre. 

3. Taking A as the centre and AB as the radius ,draw an arc at point B 

4. Taking AC as the radius and B as the centre ,draw an arc above B. 

5. Taking AB as the radius and and C as the centre ,draw an arc to meet the previous arc at D. 

6. Join the intersection of the arcs at point D to C. 

7. The line segment AB is parallel to CD 

5.4 Construction of special angles 

• In construction, of angles, we use a ruler, pencil and pair of compasses only. 

• The angle bisector method can be used to create other angles. Thus, an angle of 300 is obtained by bisecting an angle of 600. 

• The supplementary angle construction method can be used to get obtuse angles. Thus, an angle of 1200 is obtained by constructing an angle of 600. 

Activity : Construction of special angles. 

Using a pencil, ruler and pair of compasses only, construct the following angles: 

5.2 Exercise Set 

5.5 Describing a Locus. 

Locus of a point is the path which it describes as it moves. 

Activity : Discovering what Locus is 

EXAMPLE 1:

EXAMPLE 2

Activity : Sketching and Describing Loci. 

Sketch and describe what happens about the following: 

1. A mark on the floor as the door opens and closes. 

2. The centre of a bicycle wheel as the bicycle travels along a straight line. 

3. A man is walking and keeping the same distance from two trees P and Q. 

4. A student is walking in the class keeping the same distance from two opposite walls. 

5.6 Relating Lines and Angles to Loci 

• locus is a set of points which satisfies a certain condition. 

Activity : Demonstration of some simple Loci 

1. Demonstrate how one can walk the same distance from a given point. 2. How one can walk the same distance from two xed points. 

3. How one can walk the same distance from a line. 

4. How one can walk the same distance from two intersecting lines 

5.7 Construction of Loci 

Activity : Construction of a locus at a point equidistant from a fixed point. 

• The locus of points that are at a constant distance from a fixed point is a circle with radius equals to constant distance. 

EXAMPLE 

Construct the locus of a point Q at a constant distance of 2 cm from a fixed point P. 

Activity : Construction of a locus of a point equidistant from a given line. 

• The locus of points that are at a constant distance from a straight line is a pair of parallel lines at a constant distance from the given straight line. 

EXAMPLE 

Construct the locus of a point P that moves a constant distance of 2 cm from a straight line AB 

NOTE:

• The locus of points at a fixed distance, d, from the point, P is a circle with the given point P as its center and d as its radius. i.e Given a fixed point, the locus of points is a circle. 

• The locus of the points at a fixed distance, d, from a line with end points AB, is a pair of parallel lines at a distance, d(apart) from AB and on either side of AB. i.e Given a straight line, the locus of points is two parallel lines. 

• The locus of points equidistant from two given points, A and B, is the perpendicular bisector of the line segment that joins the two points. i.e Given two points, the locus of points is a straight line midway between the two points. 

• The locus of points equidistant from two intersecting lines, L1 and L2, is a pair of bisectors that bisect the angles formed by line L1 and L2.i.e Given two intersecting lines, the locus of points is a pair of lines that cut the intersecting lines in half.

5.3 Exercise Set 

1. Construct the locus of a point Q that moves a constant distance of 3 cm from a straight line XY 

2. A dog is on a lead tethered to a post in the corner of a garden. The lead is 5 cm long. Describe the locus of the dog with a sketch. 

3. Construct the locus of a point equidistant from two intersecting lines. 

5.8 Construction of Geometric Figure. 

• Construction of figures is an application of the locus, since during inscribing and circumscribing we use the knowledge of angle bisector. 

Activity : Construction of geometrical figures

 Steps for circumscribing a circle on a triangle.

• Construct the perpendicular bisector of one side of triangle. 

• Construct the perpendicular bisector of another side . 

• Where they cross is the center of the Circumscribed circle. 

• Place the compass on the center point, adjust its length to reach any vertex of the triangle, and draw your Circumscribed circle 

EXAMPLE 

Steps for inscribing a circle in a triangle. 

• Bisect two angles of a triangle.

• The angle bisector will intersect at the incenter(centre point).

• Construct a perpendicular from the centre point to one side of the triangle. 

• Place the compass at the centre point and adjust its length up to where the perpendicular crosses the triangle, and then draw the inscribed circle. 

EXAMPLE

5.4 Exercise Set