Construction of Triangles (Special Cases)
Construct a triangle when a base side, a base angle and the sum of two sides are given
Say: Base (BC) A base angle (∠B)
Sum of the other two sides (AB + AC = S)
Steps for Construction
Draw a line BC of the given length using a ruler.
At B, use a protractor or compass to construct ∠B as given. Draw a ray BX extending from B along the angle.
Set the compass to the length of AB+AC=S.
Place the compass at B and draw an arc on BX.
Label the intersection point as D.
Join D to C.
Locate Point A. The perpendicular bisector of DC will intersect BX at point A. This ensures that A is positioned correctly so that AB + AC = S.
Join A to B and A to C.
ΔABC is the required triangle.
Construct a triangle when a base side, a base angle and the difference between two sides
Say: Base (BC) A base angle (∠B) Difference between the other two sides (AB - AC = d or AC - AB = d)
Steps for Construction:
Draw a line BC of the given length using a ruler.
At B, use a protractor or compass to construct ∠B as given. Draw a ray BX extending from B along the angle.
Identify whether AB > AC or AC > AB: If AB > AC, measure d on BX above B and mark point D. If AC > AB, measure d below B and mark point D.
Join D to C. Construct the perpendicular bisector of DC.
The perpendicular bisector of DC will intersect BX at point A. This ensures that AB - AC = d (or AC - AB = d).
Join A to B and A to C.
ΔABC is the required triangle.
Construct a triangle when the perimeter and two base angles are given
Say: P = AB + BC + CA Two base angles ∠B and ∠C
Steps for Construction:
Draw a horizontal line (long enough to fit the triangle). This line will temporarily represent the perimeter.
Choose any point P on this base line. P will later help define the perimeter constraint.
At P, use a protractor to construct ∠B on one side and ∠C on the other side. Draw two rays extending from P based on these angles.
Set your compass to the given perimeter (P). Place the compass at P and draw an arc that cuts both rays at two points A and C. These two points define the triangle sides AB and AC.
Join A to C to complete ΔABC.
Construct a circle segment given a chord and an angle
A circular segment is a region of a circle bounded by a chord and the corresponding arc. In this construction, we are given:
Say: A chord (AB) An angle (θ) subtended by the arc at any point on the segment
Steps for Construction:
Using a ruler, draw a straight line segment AB of the given length. This will act as the chord of the required circular segment.
Choose a point P where the arc should pass. At P, construct ∠APB = θ using a protractor.
Find the midpoint of AB. Using a compass, draw the perpendicular bisector of AB. This bisector will pass through the center O of the required circle.
Extend the perpendicular bisector until it meets the line OP at some point O. O is the center of the circle.
With O as the center and OA (or OB) as the radius, draw the complete circle using a compass.
Highlight the arc APB that corresponds to the given angle θ. The region between arc APB and chord AB is the required circular segment.