Key Concepts:
Basic Tools for Construction:
Ruler:
Used to draw straight lines and measure distances.Compass:
Used to draw circles and arcs and to measure distances that can be transferred onto diagrams.Constructing a Perpendicular Bisector:
The perpendicular bisector of a line segment divides it into two equal parts and forms a \(90^\circ\) angle with the line segment.Bisecting an Angle:
The angle bisector divides an angle into two equal parts.Constructing a Triangle:
A triangle can be constructed given the following:Constructing Loci:
Using a compass and ruler, loci of points can be constructed based on specific geometric conditions, such as points equidistant from a fixed point or line.Examples:
Example 1: Construct the perpendicular bisector of a line segment \(AB\).
Solution:
1. Place the compass at \(A\) and draw an arc above and below the line segment. \\ 2. Without changing the compass width, place the compass at \(B\) and draw arcs above and below the line segment, intersecting the first arcs. \\ 3. Use the ruler to draw a straight line through the points of intersection. \\ The resulting line is the perpendicular bisector of \(AB\).Example 2: Bisect a \(60^\circ\) angle.
Solution:
1. Draw a \(60^\circ\) angle using a protractor. Label the vertex \(O\). \\ 2. Place the compass at \(O\) and draw an arc intersecting both arms of the angle. Label the points of intersection \(A\) and \(B\). \\ 3. Place the compass at \(A\) and \(B\) and draw two arcs that intersect each other inside the angle. Label the point of intersection \(P\). \\ 4. Draw a straight line from \(O\) to \(P\). \\ The line \(OP\) bisects the \(60^\circ\) angle into two \(30^\circ\) angles.Example 3: Construct a triangle given sides \(AB = 5 \, \text{cm}\), \(AC = 4 \, \text{cm}\), and \(BC = 6 \, \text{cm}\).
Solution:
1. Draw a base line \(BC = 6 \, \text{cm}\). \\ 2. Place the compass at \(B\) and draw an arc of radius \(5 \, \text{cm}\). \\ 3. Place the compass at \(C\) and draw an arc of radius \(4 \, \text{cm}\). \\ 4. Label the point of intersection of the arcs as \(A\). \\ 5. Join \(A\) to \(B\) and \(A\) to \(C\) with straight lines. \\ The resulting triangle \(ABC\) satisfies the given dimensions.Example 4: Construct the locus of points \(3 \, \text{cm}\) away from a given line \(L\).
Solution:
1. Place the compass at a point on \(L\) and draw arcs of radius \(3 \, \text{cm}\) on both sides of the line. \\ 2. Repeat the process at multiple points along \(L\). \\ 3. Use the ruler to draw two parallel lines through the arc intersections. \\ The resulting lines are the locus of points \(3 \, \text{cm}\) away from \(L\).