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In ΔABC, the bisector of ∠A meets BC at D. If AB = 7.5 cm, BD = 5 cm and DC = 6 cm, find AC.

Class 10Triangles (Similarity)Geometry — Similarity
  • A6.25 cm
  • B9 cm
  • C7.5 cm
  • D11 cm

Step-by-Step Solution

Drawing a line through parallel to and applying BPT gives the angle-bisector result . So and cm. 6.25 cm comes from inverting the ratio; 7.5 cm assumes a bisector from A must make , which is true only in an isosceles triangle; 11 cm is , the wrong side altogether.
Answer: B) 9 cm

Key idea: Three similarity criteria

**AA (Angle-Angle)**: Two angles equal → similar. (Third angle is automatic by 180° angle sum.)

**SSS (Side-Side-Side proportionality)**: All three pairs of corresponding sides in same ratio → similar.

**SAS (Side-Angle-Side)**: Two pairs of sides proportional AND included angles equal → similar.

**Important**: SSA does NOT prove similarity (just like for congruence).

AA is by far the most-used in problems: spot two equal angles (often via parallel lines or shared angle) and declare similarity.

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