In a quadrilateral PQRS,PQRS,PQRS, if PQ∥RS,∠S=2∠Q,PS=qPQ∥RS,∠S=2∠Q,PS=qPQ∥RS,∠S=2∠Q,PS=q and RS=p.RS=p.RS=p. Find the length of the side PQ.PQ.PQ.
Answer:
p+qp+qp+q
- Let us first draw the quadrilateral PQRSPQRSPQRS.
So, ∠S=2x∠S=2x∠S=2x
Let us join RRR to a point EEE on the side PQPQPQ such that PERSPERSPERS is a parallelogram. - We know that opposite sides of a parallelogram are equal.
So, ∠PSR=∠PER=2x…(i)∠PSR=∠PER=2x…(i)∠PSR=∠PER=2x…(i) - Also, [Math Processing Error]
- The sum of angles of a triangle is 180∘.180∘.
In △ERQ△ERQ [Math Processing Error] - In △ERQ,△ERQ, [Math Processing Error]
- We are given that RS=pRS=p and PS=q.PS=q.
As, opposite sides of a parallelogram are equal,
RS=PE=pRS=PE=p
and PS=ER=qPS=ER=q
⟹EQ=q…[From (ii)]⟹EQ=q…[From (ii)] - We can see that PQ=PE+EQ=p+qPQ=PE+EQ=p+q
- Thus, PQ=p+qPQ=p+q