Question

Draw a line $l$ and a point $X$ on it. Through $X$, draw a line segment $\overline{XY}$ perpendicular to $l$. Now draw a perpendicular to $\overline{XY}$ to $Y$. (use ruler and compasses)

Answer 1

Steps of construction:
(i) Draw a line '$l$' and take point $X$ on it.
(ii) With $X$ as the centre and a convenient radius, draw an arc intersecting the line '$l$' at two points $A$ and $B$.
(iii) With $A$ and $B$ as centres and a radius greater than $XA$, draw two arcs, which cut each other at $C$.
(iv) Join $AC$ and produce it to $Y$. Then $XY$ is perpendicular to '$l$'.

(v) With $Y$ as centre and a convenient radius, draw an arc intersecting $XY$ at two points $C$ and $D$.
(vi) With $C$ and $D$ as centres and radius greater than $YD$, draw two arcs which cut each other at $F$.
(vii) Join $YF$, then $YF$ is perpendicular to $XY$ at $Y$.

Hence, we get the figure as shown below: