## Problem 4.5 Projection of Straight Lines – A line PQR, 80 mm long, is inclined to H.P. by 30^{0 }and V.P. by 45^{0}. PQ:QR :1:3. Point Q is in V.P. and 20 mm above H.P. Draw the projection of the line PQR when the point R is in the 1^{st }quadrant. Find the position of the point P.

**Procedure: **

**Procedure:**

* Step-1* Draw a horizontal line, which is x-y line of some suitable length.

* Step-2* Mark a point q’ at the distance 20 mm from the x-y line. From the point q’ draw a horizontal line which is parallel to the line x-y.

* Step-3* The true length of the line PQR is 80 mm total. But the ration of the distance of the points PR:QR is 1:3. So the distance between the points P to Q is 20 mm & Q to R is 60 mm.

* Step-4* From the point q’ draw a line of the length 60 mm up to the point r’

_{1}& a line of length 20 mm up to the point p’

_{1}such that the line should be inclined with the line passing through the point q’ at the angle 30° as shown into the figure. It is represented by an angle θ.

* Step-5* From the point q’ draw a vertical downward line up to the line x-y. And mark that point as q as shown into the figure.

* Step-6* From the point q draw a line of the length 60 mm up to the point r

_{2}& a line of length 20 mm up to the point p

_{2 }such that the line should be inclined with the line passing through the point q at the angle 45° as shown into the figure. It is represented by an angle Φ.

* Step-7* From the point r

_{2}draw a vertical line up to the line passing from the point q’. Now with q’ as center and the radius equal to the distance between the point of intersection of the previously drawn vertical line, draw an arc which will intersect with the line passing through the point r’

_{1}. And give the name of that point as point r’.

* Step-8* From the point p’

_{1}, draw a horizontal line. And draw a line p’-q’-r’ of the length 80 mm as shown into the figure. This is the elevation of the line PQR & find its inclination with the x-y line. It is represented by an angle α as shown into the figure.

* Step-9* From the point r’

_{1}draw a vertical line up to the line x-y. Now with q as center and the radius equal to the distance between the point of intersection of the previously drawn vertical line, draw an arc which will intersect with the line passing through the point r

_{2}. And give the name of that point as point r.

* Step-10* From the point p

_{2}, draw a horizontal line. And draw a line p-q-r of the length 80 mm as shown into the figure. This is the plan of the line PQR & find its inclination with the x-y line. It is represented by an angle β as shown into the figure.

* Step-11* Give the dimensions by any one method of dimensions and give the notations as shown into the figure. And make a list of the True Length & Angles made by the line CD as shown into the figure.