In the two triangles, ∠L = ∠S = 60°, with the sides marked. Find TU.

Class 10Triangles (Similarity)Geometry — Similarity

A9.5 cm
B4.67 cm
C10.5 cm
D12 cm

Step-by-Step Solution

\frac{ST}{L M} = \frac{7.5}{5} = 1.5

and

\frac{S U}{L N} = \frac{12}{8} = 1.5

, and the included angles are equal, so △

L MN

∼ △

ST U

(SAS). Hence

T U = 1.5 \times MN = 1.5 \times 7 = 10.5

cm.

Answer: C) 10.5 cm

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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