Triangle Similarity Test

To determine if \(\Delta\text{ABC}\) and \(\Delta\text{PQR}\) are similar, we check if their corresponding sides are in proportion using the SSS Similarity criterion.

Given side lengths:

• \(\Delta\text{ABC}\): \(\text{AB} = 2\text{cm}\), \(\text{BC} = 4\text{cm}\), \(\text{AC} = 5\text{ cm}\)
• \(\Delta\text{PQR}\): \(\text{PQ} = 12\text{cm}\), \(\text{QR} = 24\text{ cm}\), \(\text{PR} = 30\text{ cm}\)

Proportional Sides Check

Compare the ratios of corresponding sides. The correspondence is inferred from the order: A \(\leftrightarrow\) P, B \(\leftrightarrow\) Q, C \(\leftrightarrow\) R.

• Ratio of sides PQ and AB: \(\frac{\text{PQ}}{\text{AB}} = \frac{12}{2} = 6\)
• Ratio of sides QR and BC: \(\frac{\text{QR}}{\text{BC}} = \frac{24}{4} = 6\)
• Ratio of sides PR and AC: \(\frac{\text{PR}}{\text{AC}} = \frac{30}{5} = 6\)

All three ratios are equal to 6. This indicates that the sides are proportional.

Triangle Relationship Conclusion

Since the ratios of all corresponding sides are equal (\(\frac{\text{PQ}}{\text{AB}} = \frac{\text{QR}}{\text{BC}} = \frac{\text{PR}}{\text{AC}} = 6\)), \(\Delta\text{ABC}\) and \(\Delta\text{PQR}\) are Similar by the SSS Similarity criterion.