# Ex.10.1 Q3 Practical Geometry Solution- NCERT Maths Class 7

## Question

Let \(l\) be a line and\(\,P\,\)be a point not on \(l\). Through \(P ,\) draw a line\(\,m\,\)parallel to \(l\). Now join\(\,P\,\)to any point \(Q\) on \(l\). Choose any other point \(R\) on\(\, m \,\). Through \(\,R,\,\) draw a line parallel to \(PQ\). Let this meet \(l\) at \(S.\) What shape do the two sets of parallel lines enclose?

## Text Solution

**What is known?**

A line \(l\) and a point \(P\) not on \(l\) and line m parallel to \(l\)..

**What is unknown?**

Shape formed by two sets of two parallel lines.

**Reasoning:**

As line \(l\) is given and\(\,P\,\)is a point not on \(l\). Through \(P,\) draw a line\(\,m\,\)parallel to \(l\). Now join\(\,P\,\)to any point \(Q\) on \(l\). Choose any other point \(R\) on\(\, m \,\). Through \(\,R,\,\) draw a line parallel to \(PQ.\) Let this meet \(l\) at \(S.\)

**Steps:**

__Steps of construction –__

- Draw a line \(l\), take a point \(A\) on it. Take a point\(\,P\,\)not on \(l\) and join \(A\) to \(P.\)
- Taking \(A\) as centre and with a convenient radius draw an arc cutting \(l\) at \(B\) and \(AP\) at \(C.\)
- Taking\(\,P\,\)as the centre and with the same radius as before, draw an arc \(DE\) to intersect \(AP\) at \(F.\)
- Adjust the compasses up to the length of \(BC.\) Without changing the opening of compasses and taking \(F\) as the centre, draw an arc to intersect the previous drawn arc \(DE\) at point \(G.\)
- Join\(\,P\,\)to any point \(G\) to draw line\(\, m \,\). Line\(\,m\,\)will be parallel to \(l\).
- Join\(\,P\,\)to any point \(Q\) on line \(l\). Choose another point \(R\) on line\(\, m \,\). Similarly, a line can be drawn through point \(R\) and parallel to \(PQ.\)
- Let it meet line \(l\) at point \(S.\)

In quadrilateral \(PQSR\), opposite lines are parallel to each other.

Thus, \(PQSR\) is a parallelogram.