Given:
In a parallelogram [tex]PRTV, VQ\perp PR,VS\perp RT[/tex].
To prove:
[tex]\dfrac{PQ}{ST}=\dfrac{PV}{VT}[/tex]
Solution:
It is given that [tex]VQ\perp PR,VS\perp RT[/tex], it means [tex]\angle PQV[/tex] and [tex]\angle TSV[/tex] are right angle triangles.
In triangle PQV and triangle TSV,
[tex]\angle PQV\cong \angle TSV[/tex] (Right angles)
[tex]\angle QPV\cong \angle STV[/tex] (Opposite angles of a parallelogram)
Two corresponding angles are congruent. So, by AA property of similarity, we get
[tex]\Delta PQV\sim \Delta TSV[/tex]
We know that the corresponding parts of similar triangles are proportional. So,
[tex]\dfrac{PQ}{TS}=\dfrac{PV}{TV}[/tex]
It can be rewritten as:
[tex]\dfrac{PQ}{ST}=\dfrac{PV}{VT}[/tex]
Hence proved.
