Question

line rs intersects triangle bcd at two points and is parallel to segment dc.
which statements are correct? choose three correct answers.

• 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)\\)
• \\(\triangle bcd\\) is similar to \\(\triangle bsr\\)
• \\(\frac{br}{rd} = \frac{bs}{sc}\\)

Explanation:

Step1: Analyze the first statement

Given \( \frac{BR}{BD}=\frac{2}{3} \), by the Basic Proportionality Theorem (Thales' theorem), \( \frac{BS}{BC}=\frac{BR}{BD}=\frac{2}{3} \). If \( BS = 6 \), then \( \frac{6}{BC}=\frac{2}{3}\Rightarrow BC = 9 \), not 3. So first statement is wrong.

Step2: Analyze the second statement

By Thales' theorem, \( \triangle BSR\sim\triangle BCD \), so \( \frac{BR}{BD}=\frac{BS}{BC} \), cross - multiplying gives \( (BR)(SC)=(RD)(BS) \) (since \( BC=BS + SC \) and \( BD = BR+RD \), and from \( \frac{BR}{BD}=\frac{BS}{BC}\Rightarrow BR\times BC=BS\times BD\Rightarrow BR\times(BS + SC)=BS\times(BR + RD)\Rightarrow BR\times SC=BS\times RD \)). So second statement is correct.

Step3: Analyze the third statement

Since \( RS\parallel DC \), \( \angle BSR=\angle BCD \) and \( \angle BRS=\angle BDC \) (corresponding angles), and \( \angle B \) is common. By AA (Angle - Angle) similarity criterion, \( \triangle BCD\sim\triangle BSR \). So third statement is correct.

Step4: Analyze the fourth statement

From Thales' theorem, \( \frac{BR}{RD}=\frac{BS}{SC} \) (because \( \frac{BR}{BD - BR}=\frac{BS}{BC - BS}\), and from \( \frac{BR}{BD}=\frac{BS}{BC}=k\), \( BR = kBD \), \( BS=kBC \), then \( \frac{kBD}{BD - kBD}=\frac{kBC}{BC - kBC}\Rightarrow\frac{k}{1 - k}=\frac{k}{1 - k}\), and also from the proportion \( \frac{BR}{RD}=\frac{BS}{SC} \)). So fourth statement is correct.

Answer:

The correct statements are:

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