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: Analyze Similar Triangles

Since \( RS \parallel DC \), by the Basic Proportionality Theorem (Thales' theorem) or AA (Angle-Angle) similarity criterion, \( \triangle BCD \sim \triangle BSR \) (corresponding angles are equal as \( RS \parallel DC \), so \( \angle B \) is common, and \( \angle BRS = \angle BDC \), \( \angle BSR = \angle BCD \)). So the first statement is correct.

Step2: Check Proportional Segments

From the similarity of \( \triangle BCD \) and \( \triangle BSR \), we have \( \frac{BR}{BD}=\frac{BS}{BC} \), which can be rearranged using the property of proportions. Cross - multiplying the proportion \( \frac{BR}{RD}=\frac{BS}{SC} \) (derived from the basic proportionality theorem as \( RS \parallel DC \)) gives \( (BR)(SC)=(RD)(BS) \), so the second and fourth statements are related. Let's check the second statement: \( \frac{BR}{RD}=\frac{BS}{SC} \) is a direct result of the basic proportionality theorem (Thales' theorem) when a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides those sides proportionally. So \( \frac{BR}{RD}=\frac{BS}{SC} \) is correct.

Step3: Analyze the Third Statement

If \( \frac{BR}{BD}=\frac{2}{3} \), then from \( \triangle BCD \sim \triangle BSR \), \( \frac{BS}{BC}=\frac{BR}{BD}=\frac{2}{3} \). If \( BS = 6 \), then \( \frac{6}{BC}=\frac{2}{3}\Rightarrow BC=\frac{6\times3}{2}=9
eq3 \). So the third statement is incorrect.

Step4: Analyze the Fifth Statement

The proportion \( \frac{BR}{RS}=\frac{BS}{SC} \) does not follow from the basic proportionality theorem or the similarity of triangles. The correct proportion related to the sides is \( \frac{BR}{BD}=\frac{BS}{BC}=\frac{RS}{DC} \), so the fifth statement is incorrect.

Step5: Confirm the Fourth Statement

From \( \frac{BR}{RD}=\frac{BS}{SC} \), cross - multiplying gives \( (BR)(SC)=(RD)(BS) \), so the fourth statement is correct.

Answer:

\( \triangle BCD \) is similar to \( \triangle BSR \), \( \frac{BR}{RD}=\frac{BS}{SC} \), \( (BR)(SC)=(RD)(BS) \) (i.e., the first, second, and fourth options).