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📑 In This Chapter Guide (Table of Contents)
1. Basic Proportionality Theorem (Thales Theorem) Full Proof
Statement: 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.
Given: In ΔABC, a line DE is drawn parallel to BC, intersecting AB at D and AC at E (DE || BC).
To Prove: AD / DB = AE / EC
Construction: Join BE and CD. Draw perpendiculars DM ⊥ AC and EN ⊥ AB.
Proof:
1. Area of ΔADE = ½ × Base × Height = ½ × AD × EN
2. Area of ΔBDE = ½ × DB × EN (EN is height for obtuse angle)
3. Ratio: ar(ΔADE) / ar(ΔBDE) = (½ × AD × EN) / (½ × DB × EN) = AD / DB ... (Equation 1)
4. Similarly, considering AC as base:
ar(ΔADE) / ar(ΔCDE) = (½ × AE × DM) / (½ × EC × DM) = AE / EC ... (Equation 2)
5. Notice that ΔBDE and ΔCDE share the same base DE and lie between the same parallel lines DE and BC.
Therefore: ar(ΔBDE) = ar(ΔCDE) ... (Equation 3)
6. From Equations 1, 2, and 3, the left-hand ratios are identical. Hence:
AD / DB = AE / EC (Hence Proved)
2. Criteria for Similarity of Triangles (AAA, SSS, SAS)
Two triangles ΔABC and ΔDEF are similar (written as ΔABC ~ ΔDEF) if: (1) Corresponding angles are equal, and (2) Corresponding sides are in the same ratio.
- AAA Similarity (or AA Similarity): If two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar (third angle is automatically equal by angle-sum property).
- SSS Similarity: If the three sides of one triangle are proportional to the three sides of another triangle:
AB/DE = BC/EF = AC/DF, the triangles are similar. - SAS Similarity: If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional:
AB/DE = AC/DFand∠A = ∠D, thenΔABC ~ ΔDEF.
💡 Frequently Asked Questions (FAQ)
❓ State the Converse of Basic Proportionality Theorem.
If a line divides any two sides of a triangle in the same ratio (AD/DB = AE/EC), then the line must be parallel to the third side (DE || BC).
❓ A vertical pole of length 6 m casts a shadow 4 m long on the ground, and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
Since the sun angle of elevation is identical for both objects simultaneously: ΔPole ~ ΔTower (by AA similarity). Therefore: (Height of Pole) / (Height of Tower) = (Pole Shadow) / (Tower Shadow) ⟹ 6 / H = 4 / 28 ⟹ 6 / H = 1 / 7 ⟹ H = 42 meters. The height of the tower is 42 m.
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