Question

in a triangle ABC , points X and Y are on AB And AC , respectively , such that XY is parallel to BC . Which of the two following equalities always hold? (here [PQR] denotes the area of triangle PQR).
(i) [BCX]=[BCY]
(ii) ` [ACX].[ABY]=[AXY].[ABC]`
A. Neither (i) nor (ii)
B. (i) only
C. (ii) only
D. Both (i) and (ii)

Answer 1

Correct Answer - D

Clearly ar (BCX)=ar(BCY) `{ Deltas" between parallel lines & same base "}`
`implies [BCY]=[BCY]`
(i) is true
check
(ii) ar `(Delta ACX)=(1)/(2)Ac.AX sin A `
` ar (Delta ABY )=(1)/(2) AB.AY sin A `
`ar (Delta AXY) = (1)/(2) AX.AY sin A .`
`ar (Delta ABC) =(1)/(2) AB.AC sin A.`
Clearly [ACX].[ABY]=[AXY].[ABC]`
true.