Special Constructions (Leaving Cert Mathematics): Revision Notes

Special Constructions

Overview

Special constructions involve creating specific geometric figures and points based on given parameters without relying on measurement tools like a protractor. These constructions use a compass and straight edge, following precise steps.

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Constructing an Angle of $60^\circ$ Without Using a Protractor or Set Square

Objective: To construct a $60^\circ$ angle using a compass and straight edge.

Method

1. Draw a ray $QR$
2. Place the compass at $Q$ and draw an arc intersecting $QR$ at a point $Y$.
3. Without changing the compass width, place the compass at $Y$ and draw another arc intersecting the first arc at a point $X$.
4. Draw a line from $Q$ through $X$,extending to $P$ Result: The angle $\angle CAB$ is $60^\circ$

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Constructing a Rectangle Given the Lengths of Two Sides

Objective: To construct a rectangle when the lengths of adjacent sides are known.

Method

1. Draw one side of the rectangle with the given length (base).
2. At each endpoint of the base, construct perpendicular lines using a compass and straight edge.
3. Mark the lengths of the other side on each perpendicular line.
4. Connect the ends of the marked lengths to complete the rectangle. Result: A rectangle with the given side lengths is constructed.

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Constructing a Parallelogram Given the Lengths of the Sides and the Measure of the Angles

Objective: To construct a parallelogram when the lengths of adjacent sides and one angle are known.

Method

1. Draw one side of the parallelogram with the given length (base).
2. At one endpoint, construct the given angle using a compass and straight edge.
3. Mark the length of the adjacent side along the angle's arm.
4. Repeat the same steps at the other endpoint of the base to construct the opposite side.
5. Connect the endpoints of the opposite sides to complete the parallelogram. Result: A parallelogram with the given side lengths and angle is constructed.

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Constructing the Centroid of a Triangle

• Objective: To find the centroid, the point where the medians of a triangle meet.

Method

1. Draw the triangle $\triangle ABC$
2. Locate the midpoints of each side of the triangle using a compass.
3. Draw the medians by connecting each vertex to the midpoint of the opposite side.
4. The point where all three medians intersect is the centroid. Result: The centroid divides each median into two segments in the ratio $2:1$.

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Constructing the Orthocentre of a Triangle

Objective: To find the orthocentre, the point where the altitudes of a triangle meet.

Method

1. Draw the triangle $\triangle ABC$
2. At each vertex, construct an altitude:

• Use a compass to draw arcs that intersect the opposite side (or its extension) from the vertex.
• Use these intersections to construct a perpendicular line from the vertex to the opposite side.

3. The point where all three altitudes intersect is the orthocentre. Result: The orthocentre is the common intersection of the altitudes.

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Summary

1. Angle of $60^\circ$: Constructed using intersecting arcs from a compass.
2. Rectangle Construction: Uses perpendiculars and given side lengths.
3. Parallelogram Construction: Formed with given side lengths and an angle.
4. Centroid: Found by intersecting medians; divides medians in a $2:1$ ratio.
5. Orthocentre: Found by intersecting altitudes of a triangle. These constructions are essential for mastering geometric problem-solving and creating precise diagrams.