Basic Geometric Elements: Circles and Line Segments

Construction of Circles with given Radius/Diameter

A circle is defined as the locus of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the center, and the fixed distance is called the radius.

The diameter of a circle is a line segment passing through the center and having its endpoints on the circle. The diameter is twice the length of the radius ($d = 2r$ or $r = d/2$).

Constructing circles accurately primarily requires a compass.

Construction with a Given Radius

This construction method is direct, as the radius is the fundamental distance used by a compass.

Given: A desired radius length, say $r$ units (e.g., 4 cm).

Tools: Compass, Ruler.

Steps:

1. Mark the Center: Choose a point $O$ on your paper. This point will be the center of your circle.
2. Set the Compass Width: Place the pointed end of the compass on the zero mark of a ruler. Open the compass so that the pencil tip (or the other point) is exactly at the mark corresponding to the given radius $r$ (e.g., 4 cm). The distance between the compass point and the pencil tip is now precisely equal to the desired radius, $r$.
3. Draw the Circle: Place the pointed end of the compass firmly on the center point $O$ marked in Step 1. Holding the compass handle at the top (usually between the thumb and forefinger), carefully rotate the pencil tip around the center point, keeping the compass width constant. Continue rotating until you have drawn a complete, closed curve.

The resulting figure is a circle with center $O$ and radius $r$. Every point on the curve is exactly $r$ distance away from $O$.

Construction with a Given Diameter

When the diameter is given, you first need to determine the radius, as the compass is set to the radius length.

Given: A desired diameter length, say $d$ units (e.g., 6 cm).

Tools: Compass, Ruler.

Steps:

1. Calculate Radius: The radius ($r$) is always half of the diameter ($d$). Calculate the radius using the formula:

   $\text{Radius} (r) = \frac{\text{Diameter} (d)}{2}$

   ... (1)

   For example, if $d = 6$ cm, then $r = 6 \text{ cm} / 2 = 3$ cm.
2. Mark the Center: Choose a point $O$ on your paper to be the center of the circle.
3. Set the Compass Width: Use the ruler to set the compass width equal to the calculated radius $r$ from Step 1 (e.g., 3 cm). This step is identical to Step 2 in the construction with a given radius.
4. Draw the Circle: Place the compass point on $O$ and rotate the pencil tip to draw the complete circle, keeping the width constant.

The resulting figure is a circle with center $O$ and radius $r = d/2$, which means it has the given diameter $d$.

Example

Answer:

Construction of Line Segments of given Length

A line segment is a part of a line that is bounded by two distinct endpoints. It has a finite length.

Constructing a line segment of a specific length is one of the most fundamental geometric constructions.

Construction using a Ruler

This is the most common and direct method for drawing a line segment of a specified length.

Given: A desired length, say $L$ units (e.g., 5.5 cm).

Tools: Ruler (with metric or standard markings), Pencil.

Steps:

1. Place the Ruler: Place the ruler flat on the paper where you want to draw the line segment.
2. Mark Starting Point: Mark a point, let's call it $A$, on the paper right next to the edge of the ruler, precisely at the zero mark (0 cm) on the ruler's scale.
3. Locate and Mark End Point: Keeping the ruler steady, find the mark on the ruler's scale that corresponds to the desired length $L$ (e.g., 5.5 cm). Mark a second point, $B$, on the paper right next to the edge of the ruler at this specific mark.
4. Draw the Segment: Using the ruler as a guide, draw a straight line connecting point $A$ to point $B$. Ensure the line is drawn carefully along the edge of the ruler to be straight.

The resulting figure is the line segment $AB$. By construction, its length is equal to $L$. The length of the line segment $AB$ is often denoted as $AB$ or $\overline{AB}$.

Example

Answer:

Copying a Line Segment

Copying a line segment means constructing a new line segment that has the exact same length as a given line segment. This is a fundamental skill in geometric constructions, often performed using only a compass and a straightedge (an unmarked ruler used only for drawing straight lines, not for measuring specific distances).

Construction using Compass and Straightedge

This method is crucial in classical geometry where measurement using a ruler's scale is not permitted, only drawing straight lines and arcs/circles.

Given: A line segment $AB$.

Tools: Compass, Straightedge.

Goal: Construct a line segment $PQ$ such that the length of $PQ$ is equal to the length of $AB$ ($PQ = AB$).

Steps:

1. Draw a Reference Line: Use the straightedge to draw a line $l$. This line needs to be longer than the segment you intend to copy. Choose and mark a point $P$ on this line $l$. This point $P$ will serve as one endpoint of the new line segment you are constructing.
2. Measure the Given Segment: Place the pointed end of the compass firmly on one endpoint of the given segment, say point $A$. Adjust the compass opening so that the pencil tip is placed exactly on the other endpoint, point $B$. The distance between the compass point and the pencil tip is now set to the length of the segment $AB$.
3. Transfer the Length: Without changing the compass width (this is critical for accuracy), place the pointed end of the compass on the point $P$ you marked on line $l$. Draw a small arc that intersects line $l$ at some point away from $P$. The compass ensures that every point on this arc is the same distance away from $P$ as the length $AB$.
4. Mark the Endpoint: Label the point where the arc drawn in Step 3 intersects line $l$ as point $Q$. The segment $PQ$ is now defined on line $l$.

The resulting line segment $PQ$ has the same length as the given line segment $AB$ because the compass opening, representing the distance $AB$, was precisely transferred from point $P$ to locate point $Q$.

Example

Answer: