Question

 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  

$$\frac{PQ}{SR}=\frac{AP}{AR}\left(\text{ Corresponding sides are proportional }\right)$$
$$\Rightarrow \frac{PQ}{SR}=\frac{1}{5}(AR=5AP)$$
$$\Rightarrow SR=5PQ$$ 

Hence proved.