Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Step-by-Step Solution
Key Concept: Use the definition of a tangent (a line touching a circle at exactly one point) and a secant (a line intersecting a circle at two distinct points). By constructing a line parallel to a given line, we can adjust its position so that it meets the circle either as a tangent or as a secant.
1. Draw the given line: With a ruler, draw a straight line $l$ on the paper. This line will serve as the reference for parallelism.
2. Choose a point for the centre: Select any point $O$ not on line $l$ and mark it as the centre of the required circle.
3. Draw the circle: Using a compass, place the needle at $O$, choose a convenient radius $r$, and draw the circle $C$.
4. Construct a tangent parallel to $l:
- From the centre $O$, draw a perpendicular to the given line $l$; let the foot of the perpendicular be $H$.
- Measure the distance $OH$.
- On the same side of $l$ as the circle, mark a point $P$ on the line through $H$ such that $HP = OH$ (i.e., translate the perpendicular distance to the other side of $l$). The line through $P$ parallel to $l$ will be at a distance $r$ from $O$.
- With a ruler, draw the line $t$ through $P$ parallel to $l$. Since the distance from $O$ to $t$ equals the radius $r$, $t$ touches the circle at exactly one point – it is the required tangent.
5. Construct a secant parallel to $l:
- Choose a point $Q$ on the same side of $l$ such that the perpendicular distance from $O$ to the line through $Q$ parallel to $l$ is less than $r$ (for example, take $Q$ on the line through $H$ but nearer to $l$ than $P$).
- Draw the line $s$ through $Q$ parallel to $l$. Because the distance from $O$ to $s$ is less than the radius, the line $s$ cuts the circle at two points $A$ and $B$ – hence $s$ is a secant.
6. Verification:
- Check that $t$ touches the circle at only one point (use a compass to confirm no second intersection).
- Verify that $s$ intersects the circle at two distinct points.
7. Label the figure: Mark the centre $O$, the tangent $t$, the secant $s$, the points of contact $T$ (for tangent) and $A, B$ (for secant), and indicate that $t \parallel s \parallel l$.
The construction satisfies the requirement: two lines parallel to the given line, one tangent and the other a secant to the drawn circle.
Correct Answer: A correctly drawn figure showing a circle with centre $O$, a given line $l$, a line $t$ parallel to $l$ touching the circle at exactly one point (tangent), and another line $s$ parallel to $l$ intersecting the circle at two points (secant). All constructions must be justified as described in the step‑wise solution.