Question

In the figure, given below, straight lines $AB$ and (CD) intersect at $P$, and $AC||BD$. Prove that:

ii) If $BD=2.4cm,AC=3.6cm,PD=4.0cm$ and $PB=3.2cm$; find the length of $PA$ and $PC$.

Answer 1

Step $1$: Proving $\Delta APC$ and $\Delta BPD$ are similar

Given: $AC||BD;AB$ and $CD$ intersect at $P$.


Now, in $\Delta APC$ and $\Delta BPD$
$\angle APC=\angle BPD$ [Vertically opposite angles]
And, $\angle ACP=\angle BDP$ [alternate angles]
So, by $AA$ criterion, $\Delta APC$ ~ $\Delta BPD$

Hence, $\Delta APC$ and $\Delta BPD$ are similar.

Step $2$: Finding the length of $PA$ and $PC$

Given:
$BD=2.4cm,$
$AC=3.6cm,$
$PD=4.0cm,$
$PB=3.2cm$


As, $\Delta APC$ ~ $\Delta BPD$
So, $\frac{AP}{BP}=\frac{PC}{PD}=\frac{CA}{DB}$ [$\because$ the corresponding sides of similar triangles are proportional]
$\Rightarrow \frac{AP}{3.2}=\frac{PC}{4.0}=\frac{3.6}{2.4}$
$\Rightarrow \frac{AP}{3.2}=\frac{3.6}{2.4}$
$\Rightarrow AP=4.8cm$ or $PA=4.8cm$

And $\frac{PC}{4.0}=\frac{3.6}{2.4}$
$\Rightarrow PC=6.0cm$

Hence, lengths of $PA$ and $PC$ are $4.8cm$ and $6.0cm$ respectively.