Basic Proportionality Theorem (Thales' Theorem) Verification
What you will learn
Verify Thales' Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio (AD/DB = AE/EC).
Materials needed
- Cardboard sheet with parallel ruled line grid (spaced 1 cm apart)What is it?Cardboard sheet with parallel ruled line grid (spaced 1 cm apart)₹50
Thales theorem acrylic model board with parallel sliding runner bar and engraved scales.
Specifications- Base Triangle
- Scalene triangle board AB = 240 mm, AC = 300 mm, BC = 360 mm
- Parallel Slider
- Brass runner guided in twin parallel slots
- Graduation
- Direct read millimeter scales along both legs
- Colored construction paper triangles of varying shapesWhat is it?Colored construction paper triangles of varying shapes₹95
Pair of large 30 cm technical drafting set squares (45° and 60°/30°).
Specifications- Material
- Optical acrylic with bevelled inking edge
- Size
- 300 mm hypotenuse
- 30 cm Metric Ruler (millimeter divisions)What is it?30 cm Metric Ruler (millimeter divisions)₹140
Engineering drawing paper and precision mechanical pencil.
Specifications- Paper
- Cartridge paper 200 GSM
- Pencil
- 0.5 mm Pentel drafting pencil with 2H lead
- Protractor and Pair of Set Squares setWhat is it?Protractor and Pair of Set Squares set₹185
Standard laboratory apparatus used in Basic Proportionality Theorem (Thales' Theorem) Verification.
Specifications- Category
- Mathematics Apparatus
- Grade
- CBSE / NEP 2020 Laboratory Standard
- Push pins and colored thread setWhat is it?Push pins and colored thread set₹230
Standard laboratory apparatus used in Basic Proportionality Theorem (Thales' Theorem) Verification.
Specifications- Category
- Mathematics Apparatus
- Grade
- CBSE / NEP 2020 Laboratory Standard
Tick items as you collect them. Tap + to see the product, specifications and buy options.
Stay safe
- Line DE must be strictly parallel to base BC; use paired set squares to ensure zero angular error.
- Measure segment lengths with a sharp pencil and millimeter scale at eye level.
- Avoid using tiny triangles where measurement errors become significant percentages.
- Check that points D and E lie strictly on the respective sides.
Steps
- 1Draw any scalene triangle ABC on paper with sides AB = 10 cm, AC = 12 cm, BC = 14 cm.Observation
Draw arbitrary acute triangle ΔABC on drawing paper (AB = 8 cm, AC = 10 cm, BC = 12 cm).
SafetyMark triangle vertices A, B, C clearly with fine drafting pen.
- 2Using set squares, draw line segment DE parallel to side BC, intersecting AB at point D and AC at point E.Observation
Draw line DE parallel to base BC using set squares and ruler sliding technique.
SafetyKeep base ruler firmly pressed while sliding set square to maintain strict parallelism.
- 3Measure the lengths of segments AD and DB with a precision ruler.Observation
Measure segments AD = 3.0 cm and DB = 5.0 cm on left side; compute ratio AD/DB = 3.0/5.0 = 0.600.
SafetyTake readings to nearest 0.5 mm on fine metric scale.
- 4Compute ratio 1: R1 = AD / DB.Observation
Measure segments AE = 3.75 cm and EC = 6.25 cm on right side; compute ratio AE/EC = 3.75/6.25 = 0.600.
SafetyVerify that segment sums AD + DB = AB and AE + EC = AC.
- 5Measure the lengths of segments AE and EC with the ruler.Observation
Ratios match to 3 decimal places (0.600 = 0.600), verifying Thales' Basic Proportionality Theorem.
SafetyRepeat with alternative parallel lines across different heights.
- 6Compute ratio 2: R2 = AE / EC.Observation
Demonstrate via paper cutouts that triangles ΔBDE and ΔCDE have equal area on common base DE between parallel lines.
SafetyPaste cutouts onto laboratory report sheet.
- 7Compare R1 and R2 to confirm AD/DB = AE/EC. Draw two other parallel lines at different heights to verify constancy of ratio.
Tap + on a step to see what you will observe and any safety tips.
Record your observations
| Triangle Configuration | Segment AD (cm) | Segment DB (cm) | Ratio AD / DB | Segment AE (cm) | Segment EC (cm) | Ratio AE / EC |
|---|---|---|---|---|---|---|
| Position 1 (Midline DE) | 4.0 cm | 4.0 cm | 1.000 | 5.0 cm | 5.0 cm | 1.000 (Exact match) |
| Position 2 (1 : 2 Division) | 3.0 cm | 6.0 cm | 0.500 | 3.5 cm | 7.0 cm | 0.500 (Exact match) |
| Position 3 (2 : 1 Division) | 6.0 cm | 3.0 cm | 2.000 | 7.0 cm | 3.5 cm | 2.000 (Exact match) |
| Position 4 (3 : 5 Division) | 3.0 cm | 5.0 cm | 0.600 | 4.5 cm | 7.5 cm | 0.600 (Exact match) |
| Position 5 (Arbitrary DE ∥ BC) | 2.4 cm | 5.6 cm | 0.428 | 3.0 cm | 7.0 cm | 0.428 (Exact match) |
Tap a highlighted cell to write your own observation.
Result and conclusion
Measured segments AD = 4.0 cm, DB = 6.0 cm (ratio = 0.667) and AE = 4.8 cm, EC = 7.2 cm (ratio = 0.667). Ratios AD/DB = AE/EC verified across scalene, isosceles, and right triangles.
| Observation | What it means |
|---|---|
| Parallel Line Section Ratio Equality | Line DE parallel to side BC divides sides AB and AC in strictly identical proportional ratios (AD/DB = AE/EC). |
| Area-Ratio Method Proof | Area(ΔADE) / Area(ΔBDE) = AD / DB (same height); Area(ΔADE) / Area(ΔCDE) = AE / EC; since Area(ΔBDE) = Area(ΔCDE) on same base DE, ratios are identical. |
| Converse of Thales Theorem | If a line divides two sides of a triangle in the same ratio, the line is strictly parallel to the third side. |
| Foundation of Similar Triangles | Establishes the fundamental AAA similarity criterion ΔADE ~ ΔABC with proportional side ratios AD/AB = AE/AC = DE/BC. |
Sample readings. Replace with your own results.