Section B — Chapters 23–39

Constructions II

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to bisect a given angle using a compass and straight edge.
✓By the end of this lesson students will be able to construct the perpendicular bisector of a given line segment.
✓By the end of this lesson students will be able to construct a line perpendicular to a given line, both from a point on the line and from a point not on the line.
✓By the end of this lesson students will be able to construct a line parallel to a given line through a given point.
✓By the end of this lesson students will be able to divide a line segment into 2 or 3 equal segments using geometric constructions.

Key concepts

Bisecting an Angle (Construction 10)

To bisect an angle means to divide it into two equal angles. The line that divides the angle is called the angle bisector. This construction uses a compass and straight edge to find the exact middle of an angle.

Perpendicular Bisector of a Line Segment (Construction 11)

The perpendicular bisector of a line segment is a line that cuts the segment exactly in half (bisects it) and is at right angles (perpendicular) to it. Every point on the perpendicular bisector is equidistant from the endpoints of the segment.

Perpendicular to a Line from a Point On It (Construction 12)

This construction creates a line that forms a 90-degree angle with the given line at a specific point on that line. It is useful for constructing right angles at a particular position.

Perpendicular to a Line from a Point Not On It (Construction 13)

This construction creates a line that forms a 90-degree angle with the given line and passes through a specific point that is not on the line. This is often used to find the shortest distance from a point to a line.

Parallel Line through a Given Point (Construction 14)

To construct a line parallel to a given line means to draw a new line that never intersects the original line, and passes through a specific point. One common method involves using the property of corresponding angles.

Dividing a Line Segment into Equal Segments (Construction 15)

This construction allows you to divide a line segment into any number of equal parts without measuring. To divide a line segment into two equal segments, you can use the perpendicular bisector construction (Construction 11). For dividing into three or more equal segments, a common method uses parallel lines.

Key facts to remember

1Geometric constructions use only a compass and a straight edge (an unmarked ruler).
2All construction marks (arcs, lines) must be left visible in your final answer.
3Accuracy is crucial; use a sharp pencil and precise compass settings.
4The angle bisector divides an angle into two equal parts.
5The perpendicular bisector of a line segment divides it into two equal parts and is at right angles to it.
6Every point on the perpendicular bisector is equidistant from the endpoints of the segment.
7Parallel lines never meet and maintain a constant distance from each other.

Worked examples

Example 1

Construct an angle of 60 degrees and then bisect it.

IDraw a straight line segment, and mark a point O on it.

IIPlace the compass point on O, open to any convenient radius, and draw an arc that cuts the line. Label the intersection point A.

IIIWithout changing the compass setting, place the compass point on A and draw another arc that intersects the first arc. Label this intersection point B.

IVDraw a straight line from O through B. Angle AOB is 60 degrees.

VNow, to bisect angle AOB: Place the compass point on A and draw an arc inside the angle.

VIWith the same compass setting, place the compass point on B and draw another arc that intersects the first arc. Label this intersection point C.

VIIDraw a straight line from O through C. This line OC bisects angle AOB.

Answer

A 60-degree angle is constructed and then bisected, resulting in two 30-degree angles.

The construction marks for both the 60-degree angle and its bisection must be clearly visible.

Example 2

Construct a line segment XY of length 8 cm. Then construct its perpendicular bisector.

IDraw a line segment XY that is 8 cm long.

IIPlace the compass point on X. Open the compass to more than half the length of XY (e.g., 5 cm).

IIIDraw an arc above and below the segment XY.

IVWithout changing the compass setting, place the compass point on Y.

VDraw another arc above and below the segment XY, intersecting the first two arcs. Label the intersection points P and Q.

VIDraw a straight line connecting points P and Q. This line is the perpendicular bisector of XY.

Answer

A line segment XY of 8 cm is drawn, and its perpendicular bisector is constructed, passing through the midpoint of XY and forming a 90-degree angle with XY.

Ensure the compass opening is more than half the segment length for the arcs to intersect clearly.

Example 3

Draw a line L and a point P not on L. Construct a line parallel to L passing through P.

IDraw a straight line L. Mark a point P somewhere above or below L.

IIDraw a transversal line from P, intersecting line L at a point Q. (This can be at any convenient angle).

IIIPlace the compass point at Q and draw an arc that cuts both line L and the transversal PQ. Label these points R (on L) and S (on PQ).

IVWithout changing the compass setting, place the compass point at P and draw a similar arc that cuts the transversal PQ. Label the intersection point T (on PQ).

VMeasure the distance between R and S with your compass.

VIPlace the compass point at T and draw an arc that intersects the arc drawn in step 4. Label this intersection point U.

VIIDraw a straight line through P and U. This line is parallel to line L.

Answer

A line parallel to the given line L is constructed, passing through the given point P.

This method relies on constructing equal corresponding angles. All construction arcs must be visible.

Common mistakes

✗Not leaving all construction lines and arcs visible, which can lead to loss of marks.
✗Using a ruler to measure lengths or angles instead of using the compass for transferring lengths or constructing angles.
✗Inaccurate drawing due to a blunt pencil, a loose compass, or imprecise arc drawing.
✗Changing the compass setting when it should remain the same for certain steps in a construction.
✗Not understanding the purpose of each arc or line drawn, leading to incorrect steps.

Exam tips

★Always use a sharp pencil (HB or 2H recommended) and a good quality compass and straight edge for clear and accurate drawings.
★Practice each construction multiple times until you can perform it quickly and accurately without referring to notes.
★Label all points clearly as you go, especially in multi-step constructions, to avoid confusion.
★Read the question carefully to ensure you are performing the correct construction and meeting all specified requirements.

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