Question

determine if triangle opq and triangle rst are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)
answer attempt
the triangles
are
are not
similar.

Explanation:

Step1: Identify sides of triangle OPQ

Triangle OPQ has sides \( OQ = 18 \), \( PQ = 18 \), \( OP = 27 \).

Step2: Identify sides of triangle RST

Triangle RST has sides \( ST = 54 \), \( RT = 54 \), \( RS = 81 \).

Step3: Check the ratios of corresponding sides

• Ratio of \( OQ \) to \( ST \): \( \frac{18}{54} = \frac{1}{3} \)
• Ratio of \( PQ \) to \( RT \): \( \frac{18}{54} = \frac{1}{3} \)
• Ratio of \( OP \) to \( RS \): \( \frac{27}{81} = \frac{1}{3} \)

Since all corresponding sides are in proportion (\( \frac{1}{3} \)), and we can assume the included angles (between the equal sides) are equal (as the triangles have two equal sides each, suggesting isosceles triangles, so the base angles would be equal, and the included angles for the proportional sides would match), by the Side - Side - Side (SSS) similarity criterion, the triangles are similar.

Answer:

The triangles are similar.