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In Trapezium Pqrs, Side Pq || Side Sr, Ar = 5ap, as = 5aq Then Prove That, Sr = 5pq - Geometry

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Question

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

 

 

Solution

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  

\[\frac{PQ}{SR} = \frac{AP}{AR} \left( \text{ Corresponding sides are proportional } \right)\]
\[ \Rightarrow \frac{PQ}{SR} = \frac{1}{5} \left( AR = 5AP \right)\]
\[ \Rightarrow SR = 5PQ\] 
Hence proved.

 

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Solution In Trapezium Pqrs, Side Pq || Side Sr, Ar = 5ap, as = 5aq Then Prove That, Sr = 5pq Concept: Properties of Ratios of Areas of Two Triangles.
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