Class 9 Mathematic Constructions All Formula & Concept Notes | Formula Sheet

Here's a detailed overview of the formulas and concepts related to the topic "Introduction to Constructions" in Class 9 Mathematics:

1. Basic Constructions

a. Constructing a Line Segment

• Objective: To draw a line segment of a given length.
• Steps:
  1. Draw a straight line.
  2. Use a ruler to measure the given length and mark the endpoints.

b. Constructing a Perpendicular Line from a Point

• Objective: To draw a line perpendicular to a given line from a given point.
• Steps:
  1. Place the compass at the given point and draw arcs that intersect the given line on both sides.
  2. Without changing the compass width, draw arcs from the intersection points on the line. Let these arcs intersect at a point.
  3. Draw a line through the given point and the intersection point of the arcs. This line is perpendicular to the given line.

c. Constructing an Angle of 90 Degrees

• Objective: To construct a right angle (90 degrees) at a given point.
• Steps:
  1. Draw a line segment.
  2. Use a compass to draw an arc with the center at one end of the line segment.
  3. Without changing the compass width, draw another arc from the intersection point of the first arc and the line segment.
  4. Draw a line through the point of intersection of the arcs and the end of the segment.

2. Bisectors

a. Constructing the Bisector of a Line Segment

• Objective: To divide a given line segment into two equal parts.
• Steps:
  1. Place the compass at one endpoint of the line segment and draw arcs above and below the line.
  2. Without changing the compass width, draw similar arcs from the other endpoint.
  3. Draw a line through the points where the arcs intersect. This is the perpendicular bisector.

b. Constructing the Bisector of an Angle

• Objective: To divide a given angle into two equal angles.
• Steps:
  1. Place the compass at the vertex of the angle and draw an arc that intersects both sides of the angle.
  2. With the same compass width, draw arcs from the intersection points of the first arc.
  3. Draw a line through the vertex and the intersection point of the arcs. This line is the angle bisector.

3. Basic Geometric Constructions

a. Constructing an Equilateral Triangle

• Objective: To construct a triangle where all sides are equal.
• Steps:
  1. Draw a line segment of the desired length.
  2. Using a compass, draw arcs from both endpoints of the segment with the same radius as the segment.
  3. The intersection of the arcs determines the third vertex of the triangle. Draw lines to connect the endpoints and the third vertex.

b. Constructing a Triangle Given Three Sides

• Objective: To construct a triangle when the lengths of all three sides are given.
• Steps:
  1. Draw the first side of the triangle.
  2. Set the compass to the length of the second side and draw an arc from one endpoint of the first side.
  3. Set the compass to the length of the third side and draw another arc from the other endpoint of the first side. The intersection of these arcs gives the third vertex of the triangle.

4. Constructing Special Figures

a. Constructing a Regular Hexagon

• Objective: To construct a hexagon where all sides and angles are equal.
• Steps:
  1. Draw a circle with a desired radius.
  2. Using the compass set to the radius of the circle, mark off six equal arcs along the circumference.
  3. Connect the points to form the hexagon.

5. Tools Used

• Compass: For drawing arcs and circles.
• Ruler: For drawing straight lines and measuring distances.
• Protractor: For measuring and constructing angles.

These basic construction techniques form the foundation for more advanced geometrical constructions and proofs. Practicing these will help you understand the properties and relationships between geometric shapes.