Question

Line RS intersects triangle BCD at two points and is parallel to segment DC.

Triangle B C D is cut by line R S. Line R S goes through sides D B and C B. Lines D C and R S are parallel. width=

Which statements are correct? Select three options.

△BCD is similar to △BSR.
StartFraction B R Over R D EndFraction = StartFraction B S Over S C EndFraction
If the ratio of BR to BD is Two-thirds, then it is possible that BS = 6 and BC = 3.
(BR)(SC) = (RD)(BS)
StartFraction B R Over R S EndFraction = StartFraction B S Over S C EndFraction

Answer 1

Correct choices: 1, 2 and 4.

Reasons (brief):
- 1) △BCD ~ △BSR: true because RS ∥ DC so the smaller triangle BSR is similar to BCD by AA (corresponding angles equal).
- 2) BR/RD = BS/SC: true. From similarity BR/BD = BS/BC = k. Then BR = k BD, RD = (1−k)BD, BS = k BC, SC = (1−k)BC, hence BR/RD = k/(1−k) = BS/SC.
- 4) (BR)(SC) = (RD)(BS): true. Expand SC = BC − BS and RD = BD − BR and use BR·BC = BS·BD (from similarity) to get BR·SC = RD·BS.

False statements:
- The third is impossible: if BR/BD = 2/3 then BS/BC = 2/3, so BS = 6 would force BC = 9, not 3.
- The fifth is not generally true (BR/RS = BD/DC by similarity, which need not equal BS/SC).