It is given that ΔABC ~ ΔQRP. Which of these equalities of ratios is correct?

Class 10Triangles (Similarity)Geometry — Similarity

A $\frac{A B}{PQ} = \frac{BC}{QR}$
B $\frac{A B}{QP} = \frac{BC}{PR}$
C $\frac{BC}{PQ} = \frac{C A}{QR}$
D $\frac{A B}{QR} = \frac{C A}{PQ}$

Step-by-Step Solution

The statement Δ

A BC

~ Δ

QRP

means A ↔

Q, B

↔

R, C

↔

P

. So

A B

pairs with

QR, BC

pairs with

RP

and

C A

pairs with

PQ

, giving

\frac{A B}{QR} = \frac{BC}{RP} = \frac{C A}{PQ}

. The other three options all come from reading the second triangle as "

PQR

" instead of "

QRP

" — the classic wrong-vertex-order error.

Answer: D)

\frac{A B}{QR} = \frac{C A}{PQ}

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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