Question

Given : In trapezium PQRS, side PQ || side SR, AR = 5AP, AS = 5AQ then prove that, SR = 5PQ

Answer 1

Given:
side PQ || side SR
AR = 5AP,
AS = 5AQ
To prove: SR = 5PQ
Proof: In ∆APQ and ∆ARS
∠PAQ = ∠RAS (Vertically Opposite angles)
∠PQA = ∠RSA (Alternate angles, side PQ || side SR and QS is a transversal line)
By AA test of similarity
∆APQ ~ ∆ARS
$$\begin{gathered} \frac{\mathrm{PQ}}{\mathrm{SR}}=\frac{{\displaystyle \mathrm{AP}}}{{\displaystyle \mathrm{AR}}}\left(\mathrm{Corresponding}\mathrm{sides}\mathrm{are}\mathrm{proportional}\right) \\ \Rightarrow \frac{\mathrm{PQ}}{\mathrm{SR}}=\frac{{\displaystyle 1}}{{\displaystyle 5}}\left(\mathrm{AR}=5\mathrm{AP}\right) \\ \Rightarrow \mathrm{SR}=5\mathrm{PQ} \end{gathered}$$​
Hence proved.