Constructions
Why This Chapter Matters
Constructions is a guaranteed 2-4 marks chapter in CBSE board exams — it tests your ability to draw accurate geometric figures using only a compass and straightedge. The examiner checks your construction steps AND the accuracy of your diagram. Clean, labelled construction earns full marks.
Prerequisites
•Chapter 6 — Triangles (similarity, Pythagoras)
•Chapter 10 — Circles (tangent, radius)
•Basic compass and ruler usage
Core Concepts
1. Division of a Line Segment
Construction 1: Divide a line segment AB in the ratio m:n (say 2:3)
Steps:
1.Draw line segment AB of given length
2.Draw a ray AX making any acute angle with AB
3.Along AX, mark (m+n) = 5 equal points: A₁, A₂, A₃, A₄, A₅
4.Join A₅ to B
5.Through A₂, draw a line parallel to A₅B (using compass for equal angles)
6.The line through A₂ meets AB at point C
7.C divides AB in ratio 2:3 ✓
Justification: By Basic Proportionality Theorem — the parallel line creates equal ratios.
2. Construction of Similar Triangle (Scale Factor)
Construction 2: Construct a triangle similar to △ABC with sides 3/4 of corresponding sides
Case 1: Scale factor < 1 (new triangle smaller)
Steps:
1.Draw △ABC of given measurements
2.Draw ray BX making acute angle with BC
3.Mark 4 equal points on BX: B₁, B₂, B₃, B₄
4.Join B₄ to C
5.Through B₃, draw line parallel to B₄C, meeting BC at C'
6.Through C', draw line parallel to CA, meeting BA at A'
7.△A'BC' is the required triangle (sides = 3/4 of △ABC)
Case 2: Scale factor > 1 (new triangle larger) — Mark denominator many points, join last to C, from the numerator's point draw parallel.
3. Tangent to Circle from External Point
Construction 3: Draw tangents from external point P to circle with centre O
Steps:
1.Join OP
2.Find midpoint M of OP (using perpendicular bisector construction)
3.Draw circle with centre M and radius MO (= MP)
4.This circle intersects the original circle at points Q and R
5.Join PQ and PR
6.PQ and PR are the required tangents ✓
Justification: ∠OQP = 90° (angle in semicircle with diameter OP), but ∠OQP = 90° means OQ ⊥ PQ, which means PQ is tangent to original circle at Q. □
Solved Examples
Example 1 (Most Common in Board)
Q: Construct a triangle with sides 4 cm, 5 cm, 6 cm, and then construct another triangle similar to it whose sides are 5/3 of corresponding sides.
Scale factor = 5/3 > 1, so new triangle is LARGER.
Steps:
1.Draw △ABC with AB = 4 cm, BC = 5 cm, CA = 6 cm
2.Draw ray BX at acute angle to BC on the side away from A
3.Mark 5 points B₁...B₅ on BX (for denominator = 3, mark 5 points — use max of numerator and denominator)
4.Actually: mark the larger of (5, 3) = 5 points
5.Join B₃ to C (the 3rd point = denominator)
6.Through B₅, draw line ∥ B₃C, meeting BC extended at C'
7.Through C', draw line ∥ CA, meeting BA extended at A'
8.△A'BC' is the required triangle
Example 2
Q: Draw a tangent to a circle of radius 3 cm from a point 5 cm from centre.
Length of tangent = √(5² − 3²) = √16 = 4 cm (verify after construction)
Construction steps as above:
•Draw circle radius 3 cm, centre O
•Mark point P at 5 cm from O
•Find midpoint M of OP
•Draw semicircle with M as centre, MO as radius
•Points of intersection with original circle are Q and R
•PQ and PR are tangents (verify: PQ = PR = 4 cm)
PYQs
2023
Q: Construct a triangle ABC in which BC = 7 cm, ∠B = 45°, AB = 5 cm. Draw similar triangle with scale factor 3/2.
(Construction + description of steps)
2022
Q: Draw tangents to a circle of radius 5 cm from a point 13 cm away.
(After construction, tangent length = √(169−25) = 12 cm)
2021
Q: Divide a line segment 8 cm long in ratio 3:5. Write construction steps clearly.
Revision Notes
CONSTRUCTION RULES:
1. Always write number of steps clearly
2. Show all arcs — don't erase
3. Label all points as you go
4. Verify measurement in the end
DIVISION OF AB in m:n:
Mark (m+n) points on ray from A
Join nth point to B
Through mth point, draw parallel to get division
TANGENT CONSTRUCTION:
External point → join to centre → find midpoint → draw semicircle
SIMILAR TRIANGLE (scale factor p/q):
p/q < 1 → mark q points, join qth to C, draw ∥ through pth point
p/q > 1 → mark p points, join qth to C extended, draw ∥ through pth point
Related Topics
•Chapter 10 — Circles (tangent properties)
•Chapter 6 — Triangles (BPT, similarity)
