In trapezium PQRS, side PQ || side SR, AR = 5AP, AS = 5AQ then prove that, SR = 5PQ
Practice Set 1.3 | Q 5 | Page 22 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.
Given:
PQ || SR
AR = 5AP
AS = 5AQ
To prove: SR = 5PQ
Proof:
In ∆APQ and ∆ARS, we have:
∠PAQ = ∠RAS (Vertically Opposite angles)
∠PQA = ∠RSA (Alternate angles, PQ || SR and QS is a transversal line)
By AA test of similarity, we have ∆APQ ~ ∆ARS
Therefore, we can write:
PQ/SR = AP/AR (Corresponding sides are proportional)