📚 Concepts

📚 Geometrical Construction

• The process of drawing shapes, angles, and lines accurately using only a compass and straightedge, without measurement.

📚 Constructing a Dividing Line Segment

• Dividing a given line segment in a specific ratio using only compass and ruler.

📚 Constructing Similar Triangles

• Creating a triangle similar to a given triangle using the concept of proportionality.

📚 Tangents to a Circle

• Drawing tangents to a circle from a point outside or on the circle using geometric methods.

📚 Construction Conditions

• Constructions require exact conditions such as a given base length, angles, or ratios, depending on the problem.

📚 Difference from Drawing

• Constructions use logical geometric rules and instruments only — not measurement tools like protractors or scales.

✍️ Construction Steps

✍️ Dividing a Line Segment in Ratio m:n

1. Draw the line segment AB.

2. Make an acute angle at A and mark m + n equal arcs.

3. Join the last arc to B.

4. Draw a parallel line from the mᵗʰ arc to intersect AB.

5. That point divides AB in the ratio m:n.

   
   

✍️ Constructing a Triangle Similar to a Given Triangle (Scale Factor k)

1. Draw the given triangle ABC.

2. Extend a base side and make an acute angle.

3. Mark k equal arcs (more or less based on scale).

4. Join the required arc point to vertex and draw a parallel line using compass to complete the triangle.

   
   

✍️ Constructing Tangents to a Circle from a Point Outside

1. Draw the circle with center O.

2. Join the external point P to O.

3. Find the midpoint of OP and draw a semicircle on it.

4. The points where the semicircle intersects the circle are points of tangency.

5. Join those points to P to get tangents.

   
   

✍️ Constructing Tangents from a Point on the Circle

1. Draw the radius to the point on the circle.

2. Draw a perpendicular to the radius at that point this is the tangent.