A rope is tied from the top of a 16 m flagpole to a peg in the ground 12 m from the foot of the pole. The ground is level and the rope is tight. How long is the rope?

Class 10Triangles (Similarity)Geometry — Similarity

A28 m
B20 m
C10.6 m
D14 m

Step-by-Step Solution

The pole, the ground and the rope form a right triangle, so

rope^{2} = 1 6^{2} + 1 2^{2} = 256 + 144 = 400

and the rope is 20 m.

Answer: B) 20 m

Key idea: Similarity vs congruence

**Congruent**: same shape AND same size. All corresponding parts equal.
**Similar**: same shape, possibly DIFFERENT size. Corresponding angles equal; corresponding sides PROPORTIONAL (in fixed ratio k).

If two triangles are similar with scale factor

k

:
- Corresponding sides differ by factor

k

.
- Corresponding angles are equal (always, by definition).
- Areas differ by factor

k^{2}

— critical fact, and a common test question.

**Examples**:
- Equilateral triangles are ALL similar (same angles 60°-60°-60°).
- A photograph and its enlarged print are similar.
- Two right triangles sharing one acute angle are similar (AA).

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