Question

line rs intersects triangle bcd at two points and is parallel to segment dc. which statements are correct? select three options. □ △bcd is similar to △bsr. □ \\(\frac{br}{rd} = \frac{bs}{sc}\\) □ if the ratio of br to bd is \\(\frac{2}{3}\\), then it is possible that bs = 6 and bc = 3. □ (br)(sc) = (rd)(bs) □ \\(\frac{br}{rs} = \frac{bs}{sc}\\)

Explanation:

Step1: Verify triangle similarity

Since $RS \parallel DC$, by the AA (Angle-Angle) Similarity Criterion:

• $\angle BRS = \angle BDC$ (corresponding angles)
• $\angle BSR = \angle BCD$ (corresponding angles)

Thus, $\triangle BCD \sim \triangle BSR$.

Step2: Check proportional segment ratios

From similar triangles and the Basic Proportionality Theorem (Thales' theorem), the correct segment ratio is $\frac{BR}{BD} = \frac{BS}{BC}$, which rearranges to $\frac{BR}{RD} = \frac{BS}{SC}$. Cross-multiplying gives $(BR)(SC) = (RD)(BS)$.

Step3: Analyze the ratio claim

If $\frac{BR}{BD} = \frac{2}{3}$, then $\frac{BS}{BC} = \frac{2}{3}$. If $BS=6$, solving $\frac{6}{BC} = \frac{2}{3}$ gives $BC=9$, not 3, so this statement is false.

Step4: Eliminate incorrect ratio

The ratio $\frac{BR}{RS} = \frac{BS}{SC}$ does not follow from similarity or proportionality rules, so it is false.

Answer:

• $\triangle BCD$ is similar to $\triangle BSR$.
• $\frac{BR}{RD} = \frac{BS}{SC}$
• $(BR)(SC) = (RD)(BS)$