Construction of Parallel Lines, Perpendicular Lines and Special Angles for JSS 1

Construction

Performance Objectives

By the end of the lesson, students should be able to:

1. construct parallel and perpendicular lines;
2. construct special angles: $60^\circ$ and $90^\circ$;
3. bisect a given line segment.

Introduction to Geometric Construction

Geometric construction is the process of drawing accurate geometrical shapes, lines and angles using only a ruler and a pair of compasses.

In geometry, drawing a figure is different from constructing it. Drawing allows the use of measurement tools like protractors and set-squares directly to create shapes. Construction, however, relies on the logical properties of shapes and uses only a ruler (without using its scale for measurement) and a pair of compasses to achieve perfect accuracy.

To produce clean and accurate constructions, certain instruments from a mathematical set are required. The primary tools include:

1. Ruler: Used to draw straight lines connecting two points.
2. Compasses: Used to draw circles, arcs and to transfer equal lengths.
3. Pencil: Must be sharpened to a fine point to ensure thin, precise lines.
4. Eraser: Used to clean up minor mistakes without smudging the paper.

Accuracy is highly valued in technical drawing and mathematics. A slight shift in the compass point or a blunt pencil lead can lead to errors in the final construction. Therefore, all instruments must be handled with care, and compass joints must be tight enough to prevent the legs from slipping during use.

Bisecting a Line Segment

Bisection is the division of a geometric figure, such as a line segment or an angle, into two equal parts.

To bisect a line segment means to find its exact midpoint and divide it into two equal lengths. The line that divides the segment at a right angle ($90^\circ$) is called the perpendicular bisector.

How to Bisect a Line Segment

The procedure for constructing a perpendicular bisector of a given line segment $AB$ involves the following steps:

1. Draw a straight line segment and label the endpoints $A$ and $B$.
2. Place the sharp metal point of the compasses at point $A$.
3. Adjust the compasses so that the width is clearly more than half the length of the line segment $AB$.
4. Draw an arc above the line segment and another arc below the line segment.
5. Keeping the compass width exactly the same, move the sharp metal point to point $B$.
6. Draw arcs above and below the line segment to intersect the first two arcs. Label the intersection points $P$ and $Q$.
7. Use a ruler to draw a straight line through the intersection points $P$ and $Q$.
8. Label the point where the line $PQ$ cuts the segment $AB$ as $M$. Point $M$ is the exact midpoint of $AB$, and the line $PQ$ is the perpendicular bisector of $AB$.

Worked Examples on Line Bisection

Example 1: Construct a line segment $XY = 8\text{ cm}$ and bisect it using a ruler and a pair of compasses. Measure each of the resulting segments.

Solution:

1. Use a ruler to draw a horizontal line segment $XY$ of length $8\text{ cm}$.
2. Place the compass point at $X$, open it to a radius of about $5\text{ cm}$ (which is more than half of $8\text{ cm}$), and draw arcs above and below the line.
3. With the same radius, place the compass point at $Y$ and draw arcs to intersect the previous arcs at points $C$ and $D$.
4. Draw a straight line through $C$ and $D$ to cut $XY$ at point $M$.
5. Measuring the segments with a ruler shows that $XM = 4\text{ cm}$ and $MY = 4\text{ cm}$. This confirms that the line segment has been successfully bisected into two equal halves.

Example 2: Draw a line segment $PQ$ of any convenient length. Construct its perpendicular bisector to meet $PQ$ at point $O$. Verify that the angle formed at $O$ is $90^\circ$.

Solution:

1. Draw a straight line segment $PQ$ of arbitrary length.
2. Set the compass radius to more than half of $PQ$. Place the compass point at $P$ and draw arcs above and below $PQ$.
3. Place the compass point at $Q$ with the same radius and draw arcs to intersect the first arcs at points $R$ and $S$.
4. Join $R$ and $S$ with a straight line cutting $PQ$ at $O$.
5. Place a protractor at point $O$ along the line $PQ$. The reading shows that the angle $\angle ROQ$ is exactly $90^\circ$. This verifies that the bisector is perpendicular to the line segment.

Construction of Perpendicular Lines

Perpendicular lines are two lines that meet or intersect at a right angle of $90^\circ$.

There are two common scenarios for constructing perpendicular lines:

1. Constructing a perpendicular line from a given point on the line.
2. Constructing a perpendicular line from a given point outside the line.

How to Construct a Perpendicular from a Point on a Line

The procedure to erect a perpendicular from a point $P$ lying on a given line involves the following steps:

1. Draw a straight line and mark a point $P$ on it.
2. Place the sharp point of the compasses at point $P$.
3. Using a convenient small radius, draw arcs on both sides of $P$ to cut the line at points $A$ and $B$.
4. Increase the compass radius so that it is greater than the distance $AP$.
5. Place the compass point at $A$ and draw an arc above the line.
6. Keeping the same radius, place the compass point at $B$ and draw another arc to intersect the first arc at point $Q$.
7. Use a ruler to draw a straight line from $P$ through the intersection point $Q$. The line $PQ$ is perpendicular to the original line at point $P$.

How to Construct a Perpendicular from a Point Outside a Line

The procedure to drop a perpendicular from a point $T$ lying outside a given line involves the following steps:

1. Draw a straight line and mark a point $T$ some distance above the line.
2. Place the sharp point of the compasses at point $T$.
3. Open the compasses to a radius large enough to reach past the line.
4. Draw a wide arc that cuts the line at two distinct points, $U$ and $V$.
5. Move the compass point to $U$. Using a convenient radius (more than half of $UV$), draw an arc below the line.
6. Keeping the same radius, move the compass point to $V$ and draw another arc below the line to intersect the first arc at point $W$.
7. Use a ruler to draw a straight line connecting point $T$ and point $W$. The line $TW$ is perpendicular to the original line.

Worked Examples on Perpendicular Lines

Example 1: Draw a horizontal line $AB$. Mark a point $C$ on the line. Construct a perpendicular line at $C$ and label a point $D$ on this perpendicular line such that $CD = 5\text{ cm}$.

Solution:

1. Draw a horizontal line and label it $AB$. Mark a point $C$ near the middle of the line.
2. Place the compass point at $C$ and mark equal arcs on both sides of $C$ to get points $X$ and $Y$ on the line.
3. Place the compass point at $X$ with a larger radius and draw an arc above the line. Repeat from point $Y$ with the same radius to intersect the arc at point $P$.
4. Draw a straight line from $C$ through $P$. This line is perpendicular to $AB$.
5. Use a ruler to measure $5\text{ cm}$ along this perpendicular line starting from $C$. Mark this point as $D$. The segment $CD$ is exactly $5\text{ cm}$ long and perpendicular to $AB$.

Example 2: Draw a line segment $LM = 10\text{ cm}$. Mark a point $K$ about $4\text{ cm}$ above the line segment. Construct a perpendicular line from $K$ to meet $LM$ at point $N$.

Solution:

1. Draw a line segment $LM$ of length $10\text{ cm}$ using a ruler. Mark a point $K$ roughly $4\text{ cm}$ directly above the middle of the segment.
2. Place the compass point at $K$ and open it wide enough to cut $LM$ at two points, $P$ and $Q$.
3. Place the compass point at $P$ and draw an arc below the line segment. Move the compass point to $Q$ and, with the same radius, draw an arc to intersect the first arc at point $R$.
4. Join $K$ and $R$ with a straight line using a ruler. Let this line cross $LM$ at point $N$. The line $KN$ is the required perpendicular from the external point $K$ to the line segment $LM$.

Construction of Parallel Lines

Parallel lines are lines in the same plane that are always the same distance apart and never meet, no matter how far they are extended.

Parallel lines can be constructed using a ruler and set-square, or by using a ruler and a pair of compasses. The compass method relies on constructing equal perpendicular distances from two points on a given line.

How to Construct Parallel Lines Using Compasses

The procedure to construct a line parallel to a given line $AB$ at a given distance $d$ involves the following steps:

1. Draw a straight line $AB$ and mark two widely spaced points, $P$ and $Q$, on the line.
2. Construct a perpendicular line at point $P$ and another perpendicular line at point $Q$ using the standard perpendicular construction method.
3. Set the compasses to the required parallel distance $d$ using a ruler.
4. Place the sharp point of the compasses at $P$ and draw an arc on the perpendicular line above $P$ to mark point $R$.
5. Keeping the same compass width $d$, place the sharp point at $Q$ and draw an arc on the perpendicular line above $Q$ to mark point $S$.
6. Use a ruler to draw a straight line through points $R$ and $S$. The line $RS$ is parallel to the line $AB$ at a distance of $d$.

Worked Examples on Parallel Lines

Example 1: Draw a straight line $PQ$. Construct a line $RS$ parallel to $PQ$ at a perpendicular distance of $4\text{ cm}$ above it.

Solution:

1. Draw a horizontal line and label it $PQ$. Mark two points, $X$ and $Y$, on the line about $6\text{ cm}$ apart.
2. Erect a perpendicular line at point $X$ and another perpendicular line at point $Y$.
3. Set the compasses to a radius of $4\text{ cm}$ using a ruler.
4. Place the compass point at $X$ and mark an arc on the perpendicular line above $X$ to get point $R$.
5. Place the compass point at $Y$ and mark an arc on the perpendicular line above $Y$ to get point $S$.
6. Draw a straight line through $R$ and $S$. The line $RS$ is parallel to $PQ$ at a distance of $4\text{ cm}$.

Example 2: Draw a line segment $AB = 7\text{ cm}$. Construct a parallel line segment $CD$ of the same length at a distance of $3.5\text{ cm}$ below $AB$.

Solution:

1. Draw a horizontal line segment $AB = 7\text{ cm}$.
2. Erect perpendicular lines downwards from both endpoints $A$ and $B$.
3. Set the compasses to a radius of $3.5\text{ cm}$.
4. Place the compass point at $A$ and mark an arc on the perpendicular line below $A$ to get point $C$.
5. Place the compass point at $B$ and mark an arc on the perpendicular line below $B$ to get point $D$.
6. Join points $C$ and $D$ with a straight line. The segment $CD$ is parallel to $AB$ at a distance of $3.5\text{ cm}$.

Construction of Special Angles

An angle is the measure of turn or space between two intersecting lines, meeting at a point called the vertex.

Special angles like $60^\circ$ and $90^\circ$ can be constructed accurately using only a ruler and a pair of compasses, without using a protractor.

How to Construct a $60^\circ$ Angle

The construction of a $60^\circ$ angle is based on the properties of an equilateral triangle, where all three angles are equal to $60^\circ$. The procedure involves the following steps:

1. Draw a straight horizontal line and mark a starting point $O$ on it. This point $O$ will be the vertex of the angle. Mark another point $A$ on the line to the right of $O$.
2. Place the sharp point of the compasses at $O$ and open it to a convenient radius.
3. Draw a wide arc that starts from the line $OA$ and curves upwards. Label the point where the arc cuts the line $OA$ as $P$.
4. Keeping the compass width exactly the same, move the sharp point of the compasses to point $P$.
5. Draw an arc that intersects the first wide arc. Label this intersection point as $Q$.
6. Use a ruler to draw a straight line from $O$ through the point $Q$.
7. The angle $\angle QOP$ (or $\angle QOA$) is exactly $60^\circ$.

How to Construct a $90^\circ$ Angle

A $90^\circ$ angle is constructed by erecting a perpendicular line on a straight line. The procedure involves the following steps:

1. Draw a straight horizontal line and mark a point $O$ near the middle. This point $O$ will be the vertex of the $90^\circ$ angle.
2. Place the sharp point of the compasses at $O$ and draw arcs of equal radius on both sides of $O$ to cut the line at points $X$ and $Y$.
3. Place the compass point at $X$ and open the compasses to a radius greater than the distance $OX$.
4. Draw an arc above the line.
5. Keeping the same radius, place the compass point at $Y$ and draw another arc to intersect the first arc at point $Z$.
6. Use a ruler to draw a straight line from $O$ through the intersection point $Z$.
7. The angle $\angle ZOX$ (and $\angle ZOY$) is exactly $90^\circ$.

Worked Examples on Special Angles

Example 1: Construct an angle of $60^\circ$ at point $P$ on a line segment $PQ$. Use a protractor to verify the accuracy of the construction.

Solution:

1. Draw a horizontal line segment $PQ$.
2. Place the compass point at $P$ and draw a wide arc cutting $PQ$ at point $A$.
3. With the same radius, place the compass point at $A$ and draw an arc to intersect the first arc at point $B$.
4. Draw a line from $P$ through $B$.
5. Place the center of a protractor at point $P$ along the line $PQ$. The line $PB$ aligns perfectly with the $60^\circ$ mark on the protractor scale, confirming the construction is accurate.

Example 2: Construct a $90^\circ$ angle at the endpoint $A$ of a line segment $AB = 6\text{ cm}$.

Solution:

1. Draw a line segment $AB = 6\text{ cm}$ using a ruler. Extend the line slightly to the left of $A$ using a dashed line. This extension provides space to draw arcs on both sides of $A$.
2. Place the compass point at $A$ and draw small arcs of equal radius on both sides of $A$ (one on the dashed extension and one on the segment $AB$) to get points $X$ and $Y$.
3. Place the compass point at $X$ with a radius greater than $AX$ and draw an arc above the line.
4. With the same radius, place the compass point at $Y$ and draw an arc to intersect the first arc at point $C$.
5. Draw a straight line from $A$ through $C$. The angle $\angle CAB$ is exactly $90^\circ$.

Summary of Construction Rules for Accuracy

To maintain high standards of accuracy in geometric constructions, the following rules should be observed:

1. Keep the pencil lead sharp at all times. A thick line introduces measurement errors.
2. Ensure the compass hinge is tight so that the radius does not change accidentally during an arc drawing.
3. Do not press too hard on the paper. Construction arcs should be thin and light, while the final lines should be clear and distinct.
4. Never erase construction arcs. They are proof of the geometric method used to achieve the final shape or angle.

Evaluation Exercises

The following exercises are designed to test the understanding of construction techniques. Complete each task carefully using a sharp pencil, ruler and compasses:

1. Draw a line segment $AB = 9\text{ cm}$. Construct its perpendicular bisector to meet $AB$ at point $M$. Measure the length of $AM$ and $MB$ to verify your work.
2. Draw a straight line $XY$. Mark a point $P$ on the line. Construct a perpendicular line to $XY$ passing through point $P$.
3. Draw a straight line $LM$. Mark a point $K$ about $5\text{ cm}$ above the line. Construct a perpendicular line from $K$ to meet $LM$ at point $N$.
4. Draw a horizontal line segment $PQ = 8\text{ cm}$. Construct a line $RS$ parallel to $PQ$ at a perpendicular distance of $3\text{ cm}$ above it.
5. Construct an angle of $60^\circ$ at the vertex $O$ of a line segment $OB$.
6. Construct an angle of $90^\circ$ at the vertex $A$ of a line segment $AC$.
7. Draw a line segment $EF = 10\text{ cm}$. Bisect the segment to find its midpoint $O$. At point $O$, construct a perpendicular line and mark a point $G$ on it such that $OG = 4\text{ cm}$. Join $E$ to $G$ and $F$ to $G$. Measure the lengths of $EG$ and $FG$.