Question

The following statements describe triangles ABC and PQR:
For triangle ABC: AC=2, AB=4, and BC=5.
For triangle PQR: QR=7.5, PR=3, and PQ=6.
Which statement explains why ABC and PQR are either similar or not similar.

Answer 1

To determine if triangles ABC and PQR are similar, we can use the properties of triangle similarity. One common method to check for similarity is by comparing the ratios of corresponding sides of the triangles.

Given the side lengths:

**For Triangle ABC:**
- AC = 2
- AB = 4
- BC = 5

**For Triangle PQR:**
- QR = 7.5
- PR = 3
- PQ = 6

We calculate the ratios of the corresponding sides:

1. For sides AC and QR:
\[
\frac{AC}{QR} = \frac{2}{7.5} = \frac{2}{7.5} = \frac{4}{15}
\]

2. For sides AB and PR:
\[
\frac{AB}{PR} = \frac{4}{3}
\]

3. For sides BC and PQ:
\[
\frac{BC}{PQ} = \frac{5}{6}
\]

Now, we compare these ratios:
- Ratio of AC to QR: \( \frac{4}{15} \)
- Ratio of AB to PR: \( \frac{4}{3} \)
- Ratio of BC to PQ: \( \frac{5}{6} \)

Since the ratios \( \frac{4}{15} \), \( \frac{4}{3} \), and \( \frac{5}{6} \) are not all equal, this indicates that the corresponding sides of triangles ABC and PQR are not in proportion.

Thus, we can conclude that **triangles ABC and PQR are not similar**, because the ratios of their corresponding sides are not equal.