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In trapezium ABCD, AB ∥ DC. A point E on AD and a point F on BC are joined so that EF ∥ AB. If AE = 3.5 cm, ED = 5 cm and BF = 4.2 cm, find FC.

Class 10Triangles (Similarity)Geometry — Similarity
  • A2.94 cm
  • B10.2 cm
  • C5 cm
  • D6 cm

Step-by-Step Solution

Join , meeting at . In Δ ∥ gives ; in Δ ∥ gives . So and cm. 10.2 cm comes from using the whole side cm in place of — BPT applied to the wrong pair of segments; 2.94 cm comes from inverting the ratio; 5 cm is copying .
Answer: D) 6 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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