Constructing Similar Triangle
A similar triangle has the same shape as a given triangle but a different size. The corresponding angles are equal and the corresponding sides are in a fixed ratio called the scale factor.
In Class 10, we construct a triangle similar to a given triangle using a ruler and compass. The construction uses the technique of dividing a line segment in a given ratio as a foundational step.
Two cases arise depending on the scale factor:
- Scale factor < 1: The new triangle is smaller than the given triangle.
- Scale factor > 1: The new triangle is larger than the given triangle.
What is Constructing Similar Triangle?
Definition: A triangle is similar to another triangle if their corresponding angles are equal and their corresponding sides are in the same ratio (called the scale factor).
Scale Factor:
Scale Factor = Side of new triangle / Corresponding side of given triangle
Key facts:
- If scale factor = m/n, sides of the new triangle are (m/n) times the corresponding sides of the given triangle.
- All three pairs of corresponding angles remain equal.
- The construction uses the division of a line segment in ratio m : n as a sub-step.
- The new triangle and the given triangle satisfy the AA similarity criterion.
Notation:
- If △ABC ~ △A'BC' with scale factor k, we write: A'B/AB = B'C/BC = A'C/AC = k.
- The order of vertices matters: A corresponds to A', B to B, C to C'.
- The symbol ~ denotes similarity (not congruence ≅).
Constructing Similar Triangle Formula
Scale Factor Relationship:
A'B'/AB = B'C'/BC = A'C'/AC = m/n
Where:
- △ABC is the given (original) triangle
- △A'B'C' is the constructed similar triangle
- m/n is the scale factor (given as a fraction or ratio)
Area relationship:
- Area of △A'B'C' / Area of △ABC = (m/n)²
- If scale factor is 3/2, area ratio is 9/4.
Derivation and Proof
Justification of the Construction (Scale factor m/n < 1):
- Given △ABC. We need to construct △AB'C' with scale factor m/n (where m < n).
- On ray BA, divide segment BC such that BB' : B'C is determined by the ratio.
- Specifically: mark n equal arcs on a ray from B. Call them B₁, B₂, ..., Bₙ.
- Join Bₙ to C.
- Through Bₘ, draw a line parallel to BₙC, meeting BC at C'.
- Through C', draw a line parallel to CA, meeting BA at A'.
- In △BBC', the line through Bₘ parallel to BₙC divides BB in ratio m : (n − m).
- So BC' / BC = m/n (by BPT).
- Since C'A' is parallel to CA, △A'B C' ~ △ABC (by AA similarity).
- All sides of △A'BC' are in ratio m/n to corresponding sides of △ABC.
For scale factor m/n > 1: The ray extends beyond B, and the construction produces a triangle larger than the original. The parallel line from Bₘ extends beyond C to locate C' on the extended line.
Types and Properties
Two Cases of Construction:
Case 1: Scale factor m/n < 1 (Smaller Triangle)
- The new triangle fits inside the original triangle.
- Example: Scale factor 2/3 means each side is 2/3 of the original.
- The dividing point C' lies between B and C.
Case 2: Scale factor m/n > 1 (Larger Triangle)
- The new triangle extends beyond the original triangle.
- Example: Scale factor 5/3 means each side is 5/3 of the original.
- The point C' lies on BC extended beyond C.
- The line parallel to CA is drawn from C' to meet BA extended at A'.
| Property | Scale Factor < 1 | Scale Factor > 1 |
|---|---|---|
| Size of new triangle | Smaller | Larger |
| Position of C' | Between B and C | Beyond C on BC extended |
| Position of A' | Between B and A | Beyond A on BA extended |
| Total arcs on ray | n (the larger number) | m (the larger number) |
Methods
Construction Steps — Scale Factor m/n < 1 (e.g., 3/4):
- Draw the given △ABC with known sides.
- Draw a ray BX making an acute angle with BC on the side opposite to vertex A.
- Mark n (= 4) equal arcs on BX: B₁, B₂, B₃, B₄.
- Join B₄ to C.
- Through B₃ (the m-th point), draw a line parallel to B₄C, meeting BC at C'.
- Through C', draw a line parallel to CA, meeting BA at A'.
- △A'BC' is the required triangle similar to △ABC with scale factor 3/4.
Construction Steps — Scale Factor m/n > 1 (e.g., 5/3):
- Draw the given △ABC with known sides.
- Draw a ray BX making an acute angle with BC on the side opposite to vertex A.
- Mark m (= 5) equal arcs on BX: B₁, B₂, B₃, B₄, B₅.
- Join B₃ (the n-th point) to C.
- Through B₅ (the m-th point), draw a line parallel to B₃C, meeting BC extended at C'.
- Through C', draw a line parallel to CA, meeting BA extended at A'.
- △A'BC' is the required triangle similar to △ABC with scale factor 5/3.
Common Mistakes to Avoid:
- Wrong number of arcs: For scale factor m/n, mark max(m, n) arcs — not m + n (that is for line segment division, not similarity).
- Confusing which point to join to C: For m/n < 1, join the n-th point to C. For m/n > 1, join the n-th point to C and draw parallel through the m-th.
- Not extending the sides: For scale factor > 1, BC and BA must be extended beyond C and A respectively.
- Inaccurate parallel: The parallel through C' to CA must be drawn precisely — this determines A' and the final triangle.
Verification of Construction:
- Measure all three sides of the constructed triangle.
- Divide each by the corresponding side of the original triangle.
- All three ratios should equal the scale factor m/n (within measurement tolerance of ±0.1 cm).
- Measure corresponding angles with a protractor — they should be equal.
Special Cases:
- Scale factor = 1: The new triangle is congruent to the original.
- Scale factor = 1/2: The new triangle is the medial triangle (connecting midpoints of sides).
- Scale factor = 2: The new triangle has sides twice the original — the area becomes 4 times.
Solved Examples
Example 1: Construct Similar Triangle with Scale Factor 2/3
Problem: Construct a triangle similar to △ABC with sides AB = 6 cm, BC = 7 cm, AC = 8 cm, with scale factor 2/3.
Solution:
Given:
- △ABC: AB = 6, BC = 7, AC = 8 cm
- Scale factor = 2/3 (< 1, so new triangle is smaller)
Steps:
- Draw △ABC with given measurements.
- Draw ray BX at an acute angle to BC.
- Mark 3 equal arcs on BX: B₁, B₂, B₃.
- Join B₃ to C.
- Through B₂, draw a line parallel to B₃C, meeting BC at C'.
- Through C', draw a line parallel to CA, meeting BA at A'.
Result: △A'BC' is the required triangle.
Expected measurements:
- A'B = 6 × 2/3 = 4 cm
- BC' = 7 × 2/3 ≈ 4.67 cm
- A'C' = 8 × 2/3 ≈ 5.33 cm
Example 2: Construct Similar Triangle with Scale Factor 5/3
Problem: Construct a triangle similar to △PQR with PQ = 5 cm, QR = 6 cm, PR = 7 cm, with scale factor 5/3.
Solution:
Given:
- △PQR: PQ = 5, QR = 6, PR = 7 cm
- Scale factor = 5/3 (> 1, so new triangle is larger)
Steps:
- Draw △PQR with given measurements.
- Draw ray QX at an acute angle to QR on the opposite side of P.
- Mark 5 equal arcs on QX: Q₁, Q₂, Q₃, Q₄, Q₅.
- Join Q₃ to R (since n = 3).
- Through Q₅, draw a line parallel to Q₃R, meeting QR extended at R'.
- Through R', draw a line parallel to RP, meeting QP extended at P'.
Result: △P'QR' is the required triangle.
Expected measurements:
- P'Q = 5 × 5/3 ≈ 8.33 cm
- QR' = 6 × 5/3 = 10 cm
- P'R' = 7 × 5/3 ≈ 11.67 cm
Example 3: Construct Similar Right Triangle (Scale Factor 3/4)
Problem: Construct a triangle similar to a right triangle with legs 6 cm and 8 cm, using scale factor 3/4.
Solution:
Given:
- Right △ABC: AB = 6 cm (one leg), BC = 8 cm (other leg), angle B = 90°
- Scale factor = 3/4
Steps:
- Draw BC = 8 cm. At B, construct a 90° angle. Mark A such that BA = 6 cm.
- Join AC (hypotenuse = √(36 + 64) = 10 cm).
- Draw ray BX at an acute angle to BC.
- Mark 4 equal arcs: B₁, B₂, B₃, B₄. Join B₄C.
- Through B₃, draw parallel to B₄C meeting BC at C'.
- Through C', draw parallel to CA meeting BA at A'.
Result: △A'BC' is a right triangle with legs 4.5 cm and 6 cm.
Verification:
- A'B = 6 × 3/4 = 4.5 cm
- BC' = 8 × 3/4 = 6 cm
- A'C' = 10 × 3/4 = 7.5 cm
- Angle B remains 90°. Confirmed similar.
Example 4: Equilateral Triangle with Scale Factor 3/2
Problem: Construct a triangle similar to equilateral △ABC with side 4 cm, using scale factor 3/2.
Solution:
Given:
- Equilateral △ABC: each side = 4 cm, each angle = 60°
- Scale factor = 3/2 (> 1, so larger triangle)
Steps:
- Draw equilateral △ABC with side 4 cm.
- Draw ray BX at an acute angle to BC.
- Mark 3 equal arcs: B₁, B₂, B₃ (since m = 3).
- Join B₂ to C (since n = 2).
- Through B₃, draw parallel to B₂C meeting BC extended at C'.
- Through C', draw parallel to CA meeting BA extended at A'.
Result: △A'BC' is an equilateral triangle with each side = 4 × 3/2 = 6 cm.
Example 5: Isosceles Triangle with Scale Factor 4/5
Problem: Construct a triangle similar to isosceles △ABC where AB = AC = 5 cm, BC = 6 cm, using scale factor 4/5.
Solution:
Given:
- Isosceles △ABC: AB = AC = 5 cm, BC = 6 cm
- Scale factor = 4/5 (< 1)
Steps:
- Draw △ABC with given measurements.
- Draw ray BX at an acute angle to BC.
- Mark 5 arcs: B₁ to B₅. Join B₅C.
- Through B₄, draw parallel to B₅C meeting BC at C'.
- Through C', draw parallel to CA meeting BA at A'.
Result:
- A'B = 5 × 4/5 = 4 cm
- BC' = 6 × 4/5 = 4.8 cm
- A'C' = 5 × 4/5 = 4 cm
- △A'BC' is also isosceles. Confirmed.
Example 6: NCERT Exercise 11.1 Q2 — Scale Factor 3/4 with Given Angles
Problem: Construct △ABC with BC = 6 cm, ∠B = 50°, ∠C = 60°. Then construct △A'BC' similar to △ABC with scale factor 3/4.
Solution:
Step 1 — Draw △ABC:
- Draw BC = 6 cm.
- At B, construct angle 50°. At C, construct angle 60°.
- The two rays intersect at A. △ABC is drawn. (∠A = 180° − 50° − 60° = 70°).
Step 2 — Construct similar triangle:
- Draw ray BX at acute angle to BC.
- Mark 4 arcs: B₁, B₂, B₃, B₄. Join B₄C.
- Through B₃, draw parallel to B₄C meeting BC at C'.
- Through C', draw parallel to CA meeting BA at A'.
Result: BC' = 6 × 3/4 = 4.5 cm. Angles of △A'BC' are 50°, 60°, 70° — same as △ABC.
Example 7: Scale Factor 7/4 (Larger Triangle)
Problem: △ABC has AB = 4 cm, BC = 5 cm, CA = 6 cm. Construct a similar triangle with scale factor 7/4.
Solution:
Given:
- Scale factor = 7/4 (> 1)
- m = 7, n = 4
Steps:
- Draw △ABC with given sides.
- Draw ray BX at an acute angle to BC.
- Mark 7 arcs: B₁ through B₇.
- Join B₄ to C (the n-th = 4th point).
- Through B₇, draw parallel to B₄C meeting BC extended at C'.
- Through C', draw parallel to CA meeting BA extended at A'.
Expected measurements:
- A'B = 4 × 7/4 = 7 cm
- BC' = 5 × 7/4 = 8.75 cm
- A'C' = 6 × 7/4 = 10.5 cm
Example 8: Verifying Similarity
Problem: After constructing a similar triangle with scale factor 2/5, the new sides measure 2.4 cm, 3.2 cm, and 2.8 cm. The original sides are 6 cm, 8 cm, and 7 cm. Verify similarity.
Solution:
Checking ratios:
- 2.4/6 = 0.4 = 2/5 ✓
- 3.2/8 = 0.4 = 2/5 ✓
- 2.8/7 = 0.4 = 2/5 ✓
All ratios equal 2/5.
Area ratio:
- (2/5)² = 4/25
- If original area = A, new area = 4A/25.
Answer: The triangles are similar with scale factor 2/5. Confirmed.
Example 9: Construction with Scale Factor 1 (Congruent Triangle)
Problem: What happens when the scale factor is 1/1?
Solution:
Analysis:
- Scale factor = 1 means m = n.
- The m-th point and the n-th point on the ray are the same point.
- The parallel through this point coincides with BC itself.
- So C' = C and A' = A.
Answer: A scale factor of 1 produces a congruent triangle — the new triangle is identical to the original. No enlargement or reduction occurs.
Example 10: Finding Scale Factor from Given Triangles
Problem: △ABC has sides 5, 7, 9 cm. △DEF has sides 10, 14, 18 cm. Determine if they are similar, and if so, state the scale factor.
Solution:
Checking ratios:
- 10/5 = 2
- 14/7 = 2
- 18/9 = 2
All ratios are equal.
Answer: The triangles are similar with scale factor 2 (i.e., 2/1). △DEF is an enlargement of △ABC. To construct △DEF from △ABC, use scale factor 2/1 — mark 2 arcs on the ray, join the 1st to C, draw parallel through the 2nd.
Real-World Applications
Mapmaking and Scale Drawing:
- Maps use scale factors to represent large areas on paper. A map with scale 1 : 50,000 represents every 1 cm as 500 m.
- Architectural blueprints enlarge (for detail) or reduce (for overview) using scale factors.
Photography and Image Processing:
- Enlarging or reducing photographs preserves shape (angles) while changing size — this is similarity.
- Digital zoom uses similar triangle principles.
Engineering Models:
- Scale models of buildings, bridges, and vehicles are similar figures with a fixed scale factor.
- Wind tunnel models of aircraft are geometrically similar to the actual aircraft.
Shadow Problems:
- An object and its shadow form similar triangles with the sun's rays, enabling height calculations.
Key Points to Remember
- A similar triangle has the same angles but sides in a fixed ratio (scale factor).
- Scale factor m/n < 1 produces a smaller triangle; m/n > 1 produces a larger triangle.
- The construction uses division of a line segment in ratio m : n as a sub-step.
- For scale factor < 1: mark n arcs, join the n-th to C, draw parallel through the m-th.
- For scale factor > 1: mark m arcs, join the n-th to C, draw parallel through the m-th (extends beyond C).
- The construction is justified by BPT and AA similarity criterion.
- Corresponding angles of similar triangles are always equal.
- Ratio of areas = (scale factor)².
- A scale factor of 1 gives a congruent triangle (same size).
- This construction is frequently asked in CBSE board exams for 3–5 marks.
Practice Problems
- Construct △ABC with AB = 5 cm, BC = 6 cm, CA = 7 cm. Then construct a similar triangle with scale factor 3/5.
- Draw △PQR with PQ = 4 cm, ∠P = 60°, PR = 5 cm. Construct a similar triangle with scale factor 4/3.
- Construct an equilateral triangle with side 5 cm. Then construct a similar triangle with scale factor 2/3. Measure the new side.
- △XYZ has XY = 8 cm, YZ = 6 cm, XZ = 10 cm (right triangle). Construct a similar triangle with scale factor 3/4.
- Construct a triangle with sides 4, 5, 6 cm. Then construct a similar triangle with scale factor 5/4. Measure the new sides.
- If a triangle has area 24 cm² and you construct a similar triangle with scale factor 2/3, what is the area of the new triangle?
- Construct △ABC with BC = 7 cm, ∠B = 45°, ∠C = 60°. Construct a similar triangle with scale factor 5/7.
Frequently Asked Questions
Q1. What is a scale factor in similar triangle construction?
The scale factor is the ratio of corresponding sides of the new triangle to the given triangle. A scale factor of 3/4 means each side of the new triangle is 3/4 of the corresponding side of the original.
Q2. How do I decide whether to mark m or n arcs on the ray?
Mark the larger of m and n arcs. For scale factor m/n < 1, n is larger, so mark n arcs. For m/n > 1, m is larger, so mark m arcs. This ensures the construction covers the required ratio.
Q3. Do the angles change in a similar triangle?
No. All corresponding angles remain exactly equal. Only the side lengths change by the scale factor. This is the defining property of similarity.
Q4. Why do we draw the ray on the opposite side of the vertex?
Drawing the ray on the opposite side of vertex A from base BC prevents the construction lines from overlapping with the triangle. It keeps the diagram clean and the parallel line construction accurate.
Q5. Can I construct a similar triangle without first drawing the original?
No. The construction requires the original triangle as a reference. The new triangle shares vertex B and is built by proportionally modifying sides BC and BA of the original.
Q6. How is this construction related to dividing a line segment?
The core step — marking equal arcs on a ray and drawing a parallel — is exactly the technique used to divide a line segment in a given ratio. Here it is applied to side BC to locate C', and then the parallel to CA locates A'.
Q7. What happens to the area when we use scale factor k?
The area of the similar triangle is k² times the area of the original. For scale factor 2/3, the new area is (2/3)² = 4/9 of the original area.
Q8. Is this construction important for board exams?
Yes. CBSE typically asks to construct a similar triangle with a given scale factor (like 3/4 or 5/3) along with proper steps of construction. It carries 4-5 marks and requires showing the construction lines clearly.
Q9. Can we use any vertex as the common vertex?
Yes. The NCERT method uses B as the common vertex, but you could use A or C. The construction steps remain analogous — the ray is drawn from the chosen vertex along one side.
Q10. How do I verify my construction is correct?
Measure all three sides of the new triangle and check that each is in the ratio m/n with the corresponding side of the original triangle. Also verify that corresponding angles are equal using a protractor.
