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in a triangle ABC , points X and Y a...

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)

Text Solution

Verified by Experts

The correct Answer is:
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.
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