ABCD is a parallelogram with diagonal AC. If a line XY is drawn such that XZ $∥$ AB then $\frac{B X}{X C}$ =?

$\frac{A Y}{A C}$

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$\frac{D Z}{A Z}$

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$\frac{A Z}{Z D}$

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$\frac{A C}{A Y}$

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Solution

The correct option is C
$\frac{A Z}{Z D}$

In parallelogram ABCD,
AB $∥$ CD
AB $∥$ CD $∥$ XZ

In $Δ$ ABC,
AB $∥$ XZ
$\Rightarrow$ AB $∥$ XY
$∴$ $\frac{B X}{X C}$ = $\frac{A Y}{Y C}$ ..... (1) [By BPT]

In $Δ$ ACD,
CD $∥$ YZ
$∴$ $\frac{A Y}{Y C}$ = $\frac{A Z}{Z D}$ ...... (2) [By BPT]

From (1) and (2),
$\frac{B X}{X C}$ = $\frac{A Y}{Y C}$ = $\frac{A Z}{Z D}$

$\frac{B X}{X C}$ = $\frac{A Z}{Z D}$

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