CBSE Class 9 Mathematics Constructions Notes Set 20

Perpendicular Bisector and Angle Bisector

Perpendicular bisector of AB: draw arcs of equal radius (more than half of AB) from A and B on both sides; the line through the two intersection points is the perpendicular bisector. Angle bisector: draw an arc from the vertex cutting both arms, then equal arcs from these two points (radius more than half the gap between them); join the vertex to their meeting point.

Standard Angles

60°: an arc of any radius, then an arc of the same radius from the cut point. 120°: two such steps. 90°: bisect the 120° angle's supplement or use two arcs on a straight line. 30°: bisect 60°. 45°: bisect 90°. 105°: bisect the angle between 90° and 120°. 135°: bisect the angle between 120° and 150°. 150°: 180° minus 30°. Half of any constructible angle is constructible.

Triangle with Base, Base Angle and Sum of Two Sides

Draw BC and the angle B. On the arm cut BD equal to the given sum AB + AC. Join D to C and draw the perpendicular bisector of CD to meet BD at A. Then AC = AD and AB + AC = BD.

Triangle with Base, Base Angle and Difference of Two Sides

Draw BC and angle B. Cut BD equal to the difference. If AB > AC, D lies on BA; if AB < AC, D lies on the arm produced backward. Join DC and draw its perpendicular bisector, meeting the arm at A.

Triangle with Perimeter and Two Base Angles

Draw a segment PQ equal to the perimeter, make angles equal to the given base angles at P and Q, and bisect them to meet at A. The perpendicular bisectors of AP and AQ meet PQ at B and C. Then ABC is the required triangle.

Feasibility

The sum of any two sides of a triangle must exceed the third side and the difference of two sides must be smaller than the third. Check this before constructing from sums or differences.

Other Constructions

Equilateral triangle from its altitude: draw the altitude, make 30° at its top and mirror the base. Right triangle from the hypotenuse and a side: draw a semicircle on the hypotenuse and cut an arc of the given length. Triangle from base, altitude and median: A lies on a parallel to the base at the altitude distance and on an arc about the midpoint of the base with the median as radius.